UC-NRLF 


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EBEA 


BY  THE   LATE 


|PK<>f.  OF  MATItKiTAVrCS  AND  Is'AT.  PHlIvOSOPilY  i*  T»tB  EAST   INDIA  COLL.,  H^RTFORii. 


THOMAS    ATTUNSON,   M.  A., 

LATS  nciii.LAJi.  oir  co'::.r.  en,  cot.t...  oa-MBUtdge, 


JP^wm  tfjt  jTourt^  ILon"tJon  Etiition,  iuttlj  |t>T5(t{ona  63  tfjr  ISrotfterjs  of  IJic 


R-&  1  SADl.^h  _.:i,^  I  0:FPANY,  164  WIlllAM  STREKi' 

i.  J  B  T  ( )  N  :  -  ^.  t  Z  S    I'  ■  L  D  E  li"  A  I.    S  T  P.  E  L  T . 


/^3^ 


S    © 


£i^\^nrW^ 


IN   MEMORIAM 
FLORIAN  CAJORI 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/bridgealgebraOObridrich 


** 
4 


^ftMENTARY  TREATISE 


ON 

/ 


ALGEBEA. 


BY  THE  LATE 

EEY.    B.    BKLUGE,    B.D.,    xA  E.  S., 

PBOr.  or  VATDLBMATlOfl   AND  NAT.  POTLOSOFHT  IN  TUB  EAST  INDIA  COLL.,  ITEBTFOBDw 
REVISED,   IMPROVED,   AND   SIMPLIFIED, 

By    THOMAS   ATKINSON,   M.  A., 

LATE  BOHOLAB  OF  COKP.   Cn.  COLL.,  CAMBUIBaE. 


jrom  t^e  JFourt]^  Hontjon  EtiitCon,  feit^  a"&"Dit(ons  fig  t^e  33rot6cr»  of  i^t 
Cf)ri8tian  Retools. 


NEW  YORK: 
n  &  J.  SADLIER  AND  COxAIPANY,  164  WILLIAM  STRECT. 

BOSTON:  — 128   FEDEKAL    STPwEET. 
MONTREAL,    C.    E.:  — 179    NOTRE    DAME    STREET. 

1861. 


% 


Entered,  "jf/OordLO,^  to  Act  of  Coug^ew,  'd  the  year  WW, 

By  D.  &  J.  Sadueb  and  Compant, 

Bl  Mm  Clerk's  Office  of  tbe  District  Court  ot  tne  united  Siales  foi  tho  Soudiein  iAtidak  d 

New  York. 


Uim 


stereotyped  by  Biixin  A  Bbothbss,  20  North  William  street,  N.  T. 


ADVERTISEMENT. 


The  excellence  of  "Bridge's  Algebra,"  as  an  ele- 
mentarj  treatise,  lias  long  been  well  known  and 
extensively  recognised.  In  the  Preface  to  the  Second 
Edition  the  author  expressly  states,  that  "great  pains 
were  taken  to  give  to  it  all  the  perspicuity  and  simpli- 
city which  the  subject  would  admit  of,  and  to  present 
it  in  a  form  likely  to  engage  the  attention  of  young 
persons  just  entering  on  their  mathematical  studies." 
The  design,  which  he  thus  proposed  to  himself,  was 
accomplished  with  singular  felicity, — for  not  one  of 
the  many  publications  on  Algebra,  which  have  during 
a  period  of  forty  years  issued  from  the  press,  with  the 
professed  object  of  producing  a  more  simple  and  ap- 
propriate introduction  to  the  study  of  the  science,  has 
evinced  such  merits  as  justly  entitle  it  to  be  placed- 
in  comparison  with  the  performance  of  Mr.  Bridge. 
These  publications  have  accordingly  failed  to  secure 
for  themselves  the  same  measure  of  public  approba- 
tion. 

This  small  compendium  embraces  all  which  is  com- 
prised in  the  former  large  and  expensive  editions,  that 


IV  ADVERTISEMENT. 

is  either  practically  useful  or  theoretically  valuable. 
By  introducing  Equations  and  Problems  at  the  ear- 
liest stage  possible,  a  novel  and  instructive  feature — 
which  the  editor  is  persuaded  Oannot  fail  to  excite  the 
curiosity  and  stimulate  the  ardour  of  the  young  alge- 
braist, so  as  to  induce  him  to  pursue  his  studies  with 
more  than  usual  alacrity,  intelligence,  and  success — 
has  been  given  to  the  work.  A  great  variety  of  new, 
easy,  and  interesting  problems,  which  are  not  con- 
tained in  former  editions,  have  thus  been  interspersed 
in  the  several  chapters.  Besides  these  additions  many 
alterations  have  been  made,  either  for  the  sake  of  uni- 
formity of  arrangement,  or  of  rendering  the  subject 
still  more  easy  and  accessible  to  youthful  minds.  Mr. 
Bridge,  nearly  forty  years  ago,  "was  not  without  a 
hope  that  his  '  Elementary  Treatise  on  Algebra^  would 
find  its  way  into  our  Puhlic  Schools;  where,  it  was 
very  well  known,  this  branch  of  education  was  [then] 
but  little  attended  to:"  and  it  is  now  confidently 
hoped,  that  this  new  edition  will  (in  consequence  of  its 
cheap  and  improved  form)  find  its  way  into  many 
schools,  where  this  science  is  not  yet  sufficiently  at- 
tended to,  and  thus  be  the  means  of  rendering  this 
instructive  study  a  subject  of  general  education. 

T.  A. 

Guilford,  December^  1847. 


f 

CONTENTS. 


Chapter  I.  Definitions 1 

Chap.  II.     On  the  Addition,  Subtraction,  Multiplication,  and  Divi- 
sion of  Algebraic  Quantities 6 

Addition 6 

Simple  Equations. . .  .• 9 

On  the  Solution  of  Simple  Equations 10 

Problems 14-17 

Subtraction 18 

-  On  the  Solution  of  Simple  Equations 19 

Multiplication 24 

On  the  Solution  of  Simple  Equations 28 

Problems 29-35 

Division 85 

Problems  producing  Simple  Equations 41-43 

Chap.    III.  On  Algebraic  Fractions 43 

On  the  Addition,  Subtraction,  Multiplication,  and  Divi- 
sion of  Fractions 49 

On  the  Solution  of  Simple  Equations 65  ^ 

Problems 61-66 

«  On  the  Solution  of  Simple  Equations,  containing  two  or 

more  unknown  Quantities 66 

Problems '72-'75 

Chap.  IV.  On  Involution  and  Evolution 76 

On  the  Involution  of  Numbers  and  Simple  Algebraic 

Quantities 76 

On  the  Involution  of  Compound  Algebraic  Quantities. ...     78 
On  the  Evolution  of  Algebraic  Quantity 78 


VI  CONTENTS. 

PAoa 
On  the  Investigation  of  the  Rule  for  the  Extraction  of  the 
Squai-e  Root  of  Numbers 81 

Chap.    V.     On  Quadratic  Equations 83 

On  the  Solution  of  pui'e  Quadratic  Equations 84 

On  the  Solution  of  adfected  (^adratic  Equations 86 

Problems 

On  the  Solution  of  Problems  producing  Quadratic  Equa- 
tions  92-98 

On  the  Solution  of  Quadratic  Equations  containing  two 
unknown  Quantities 98 

Chap.  VI.     On  Arithmetic,  Geometric,  and  Harmonic  Progression...   102 

Problems , .  .105-109 

On  Geometric  Progression lOQ 

On  the  Summation  of  an  infinite  Series  of  Fractions  in 
Geometric  Progi*ession ;  and  on  the  method  of  finding 

the  value  of  Ch*culating  Decimals 113 

Problem 116 

On  Harmonic  Progression ,... 11*7 

Chap.  VII.  On  Permutations  and  Combinations 119 

Appendix.   On  the  Different  Kinds  of  Numbers 125 

On  the  Four  Rules  of  Arithmetic 126 

On  the  Two  Terms  of  a  Fraction 127 

On  Ratios  and  Proportions 128 

On  the  Squares  of  Numbers  and  then*  Roots 128 

On  the  Factors  and  Submultiples  of  a  Number 130 

On  Odd  and  Even  Numbers 131 

On.  Progressions 131 

On  Divisible  Numbers  without  a  Remainder IdS 

Properties  and  various  Explanations 134 

Miscellaneous  Problems 137* 


BRIDGE'S  lL(}E:BRi.  ; 

CHAPTER    I. 

DEFINITIONS. 

1.  Algebra  is  a  general  method  of  computation,  in  which 
number  and  quantity  and  their  several  relations  are  ex^ 
pressed  by  means  of  written  signs  or  symbols.  The  symbols 
used  to  denote  numbers  or  quantities  are  the  letters  of  the 
alphabet. 

2.  Known  or  determined  quantities  are  generally  repre- 
sented by  the^r^^  letters  of  the  alphabet ;  as,  a,  6,  c,  &c. 

3.  Unknown  or  undetermined  quantities  are  usually  ex- 
pressed by  the  last  letters  of  the  alphabet ;  as,  x^  y,  ^,  &c. 

4.  The  multiples  of  quantities,  that  is,  the  number  of  times 
quantities  are  to  be  taken,  as  twice  a,  three  times  b,  jive  times 
aar,  are  expressed  by  placing  numbers  before  them-,  as  2a, 
35,  hax.  The  numbers,  2,  3,  5,  are  called  the  coefficients  of 
the  quantities,  a,  5,  ax^  respectively.  When  there  is  no 
coefficient  set  before  a  quantity,  1  is  always  imderstood  :  thus 
a  is  the  same  as  la. 

5.  The  symbol  =  (read  is  equal  to)  placed  between  two 
•quantities  means  that  the  quantities  are  equal  to  each  other. 

Thus  12  pence  =  1  shilling ;  3  added  to  5  =  8 ;  2a  added  to 
4a  =  6a.     This  symbol  is  called  the  sign  of  equality/, 

6.  The  sign  +  (read  plus)  signifies  that  the  quantities  be- 
fore which  it  is  placed  are  to  be  added.  Thus  3  +  2  is  the 
same  thing  as  5  ;  and  a  +  b  +  x,  means  the  sum  of  a,  b,  and 
Xy  whatever  be  the  values  of  a,  6,  and  x, 

^  1.  Define  Algebra.  What  symbols  are  used  to  denote  numlers  or  quan- 
tities ? — 2.  What  letters  of  the  alphabet  are  employed  to  represent  IcTwwn 
or  determined  quantities  ? — 3.  How  are  unhnown  or  undetermined  quantities 
represented  ? — 4.  What  are  coefficients  ?  When  is  the  coefficient  omitted  f 
— 6.  What  is  the  sign  of  equality  ?— 6.  What  is  the  use  of  the  sign+  ? 


2  ALGEBRA. 

7. .  The  sign.  — .  (reg4  mtnus)  signifies  that  the  quantity  to 
which  it  is  prefixed^' is  ifo  be  subtracted.  Thus  3  —  2  is  the 
same  thin^  as  1  ;  a  — ,  b  means  the  difference  of  a  and  b,  or  b 
t^ken  from;  (t ;  aird  a  -*f- J' — ar,  signifies  that  x  is  to  be  sub- 
tracted  from  me  sum  oi  a  and  b. 

8.  Quantities  which  have  the  sign  +  prefixed  to  them  are 
called  positive,  and  those  which  have  the  sign  —  set  before 
them  are  termed  negative  quantities.  When  there  is  no  sign 
before  a  quantity  +  is  understood  :  thus  a  stands  for  +  «. 

9.  The  symbol  X  (read  into)  is  the  sign  of  multiplication, 
and  signifies,  that  the  quantities  between  which  it  is  placed 
are  to  be  multiplied  together.  Thus,  6x2  means  that  6 
is  to  be  multiplied  by  2 ;  and  a  X  b  X  c,  signifies  that 
a,  b,  c,  are  to  be  multiplied  together.  In  the  place  of  this 
symbol  a  dot  or  full-point  is  often  used.  Thus,  a.  b  ,c,  means 
the  same  as  a  X  b  X  c.  The  product  of  quantities  repre- 
sented by  letters  is  usually  expressed  by  placing  the  letters 
in  close  contact,  one  after  another,  according  to  the  position 
in  w^hich  they  stand  in  the  alphabet.  Thus,  the  product  of  a 
into  b  is  denoted  by  ab  ;  of  a,  b,  and  x,  by  abx ;  and  of  3,  a,  x, 
and  y,  by  Saxg. 

10.  In  algebraical  computations  the  word  therefore  often 
occurs.  To  express  this  word  the  symbol  .*.  is  generally 
made  use  of  Thus  the  sentence  "  therefore  a  +  6  is  equal 
to  c  +  c?,"  is  expressed  by  "  .*.  a  +  6  =  c  +  d" 

EXAMPLES. 

F  %  1.  In  the  algebraical  expression,  a  +  5  —  c,  let  a  =  9, 
5  =  7,  c  =  3  ;  then 

a  +  5-c=    9  +  7--3 
=  16-3 
=  13 
Ex.  2.  In  the  expression  ax  -\-  ay  —  xy,  let  a  =  5,  a;  =  2, 
y  =  7  ;  then,  to  find  its  value,  we  have 

ax  ■\-  ay  —  xy  —    5x2  +  5x7  —  2x7 
=  10  +  35-14 
=  45-14 
=  31 

Y.  IIow  is  the  symbol  —  read  ?— 8.  What  is  meant  by  'positive  and  what 
by  negative  q_iiantities  ?— 9.  Write  down  the  sign  of  multiplication  ?  Is  any 
other  warZ:  used  to  denote  multiplication  ?  When  is  no  symbol  used?— 
10.  What  symbol  is  used  to  denote  the  word  therefore  f 


Ans, 

13. 

Ans, 

3. 

Ans, 

65. 

Ans, 

29. 

Ans, 

176. 

DEFINITIONS.  3 

Ex.  3.  If  a  =  5,  5  =  4,  c  =  3,  c?  =  2,  a;  =  1,  y  =  0,  find 
the  numerical  values  of  the  following  expressions  : 

(1.)  a-{'b  +  c  +  x. 

(2.)  a  —  b  +  c  —  x^  y. 

(3.)  ab  +  3ac  —  be  +  4ccx  -—  xy. 

(4.)  abc_  —  abd  +  bed  -—  acx, 

(5.)  Zabc  +  ^cx  —  85c/^  +  axy, 

11.  The  symbol  -r-  (read  divided  by)  is  the  s^^^  o/  c^zVi- 
5^o^l,  and  signifies  that  the  former  of  two  quantities  between 
which  it  is  placed,  is  to  be  divided  by  the  latter.  Thus, 
8  -r  2  is  equivalent  to  4.  But  this  division  is  more  simply- 
expressed   by  making   the  former  quantity  the  numerator, 

and  the  latter  the  denominator  of  a  fraction :  thus  -r-  means 

o 

a  divided  by  b,  and  is  usually,  for  the  sake  of  brevity,  read 

a  by  b, 

EXAMPLES. 

Ex.  1.  If  a  =  2,  6  =  3  ;  then,  we  find  the  values  of 
,    .  3a  _  3X2  _  ^  _  ^ 

^  '^  56 "" E~x^ ■"  15 ~ y 

\H.  8a~36       8X2-3X3       16-9        7 

Ex.  2.  If  a  =  3,  6  =  2,  c  =  1,  find  the  numerical  values  of 

,..Sa  +  c  .10 

(1.)  jr-. ^ns,  ~. 

^    ^  46  +  a  11 

,    .      ab  +  ac^bc  .       7 

^^■>  2ab-2ac  +  bc'  ^"'- 8"     - 

12.  When  a  quantity  is  multiplied  into  itself  any  number 
of  times,  the  product  is  called  a  power  of  the  quantity. 

11.  By  what  symbol  is  division  denoted?  What  is  its  name?  Is 
division  ever  expressed  in  any  other  manner  ? — 12.  What  is  meant  by  the 
power  of  a  quantity  ? 


•4  ALGEBKA. 

13.  Powers  are  usually  denoted  by  placing  above  the  quan« 
tity  to  the  right  a  small  figure^  which  mdicates  how  often  the 
quantity  is  multiplied  into  itself.     Thus, 

a     -     -    -    -     i]iQ  first  power  of  a  is  denoted  by  a  {a})» 
a  X  ct  '    -    -     the  2d  power  oAsquare  of  a     "      a^ 
a  X  «  X  «     -     the  3d  power  or  cube  oi  a         "      a^. 
a  X  «  X  «  X  «  the  4th  power  of  a  "      a*. 

The  small  figures  ^,  ^,  \  &c.,  set  over  a,  are  respectively 
called  the  index  or  exponent  of  the  corrdfeponding  power  of  a. 

14.  The  roots  of  quantities  are  the  quantities  from  which 
the  powers  are  by  successive  multiplication  produced.  Thus, 
the  root  of  the  square  number  16  is  4,  because  4  X  4  =  16, 
and  the  root  of  the  cube  number  27  is  3,  since  3x3x3  = 
27. 

15.  To  express  the  roots  of  quantities  the  symbol  -^Z,  (a 
corruption  of  r,  the  first  letter  in  the  word  radix,)  with  the 
proper  index,  is  employed.     Thus, 

V^  or  -y/cp,  expresses  the  square  root  of  a. 
^a  "     '•  "         the  cube  root  of  a. 

V«  "     "  "         the  fourth  root  of  a. 

&;c.  &c. 

EXAMPLES. 

Ex.  1.  If  a  =  S,b  =  2]  then  a^  =  3  X  3  =  9,  a^  =  3  X 
3  X  3  =  27,  ^>^  =  2  X  2  X  2  X  2  =  16. 

Ex.  2.  If  a  =  64  ;    then   ^a  =  -y/64  =  8,  V«  =  V64  = 

V4  X  4  x"4  =  4,  V«  =  v/64  =  2. 

ax"^  -4-  <^* 
Ex.  3.  In  the  expression — - ,  let  a  =  3,  6  =  5,  c  =  2, 

ox '~'~  a  "^~  c 


m 


=  6.     What  is  the  numerical  value  ? 


Here  arr'  +  5^  =  3  X  6  X  6  +  5  X  5  ==  108  +  25  =  133, 
andJa;  —  a'  —  c  =  5x6  —  3x3  —  2  =  30  —  9--2=19 

«a;2  +  ^'     _  133  _  ^ 


6:r— a^  — c        19 


18.  How  Sirepotoers  denoted? — 14.  What  are  the  roots  of  qnnntities  ? — 
15.  By  what  symbol  are  the  roots  of  quantities  expressed  ?  What  is  tho 
origin  of  this  symbol  ? 


DEFINITIONS.  5 

Ex.  4.  If  a  =  1,  6  =  3,  f  =  5,  c?  =  0,  find  the  values  of 
(1.)  a^  -f  26  —  c.  Ans.    2. 

(2.)  a^  +  Sb'  -  c\  Ans.    3. 

(3.)  a"  +  2b^  +  3c^  -f^c?'.  ^ns.  94. 

(4.)  Sa'b  -  25V  +  4c^  -  4aU  ^?i5.  19. 

(5.)  a^  +  b\  Ans.  28. 

«3        A^        c^ 
(6.)|-+|-+3-.  ^«^-51- 

Ex.  5.  Let  a  =  64,  6  =  81,  c  =  1 :   find  the  values  of 

(1.)  ^a-\-^b,  Ans,  17. 

(2.)  Va+'v/^+'v/^-  ^/i5.  18. 

(3.)  V^  ^^s.  72. 

16.  When  several  quantities  are  to  be  taken  as  one  quan- 
tity^ they  are  enclosed  in  brackets^  as  (  ),  j  (,  [  ]. 
Thus,  (a  ■\-  b  —  c)  ,{d  —  e)  signifies  that  the  quantity  repre- 
sented  by  a  +  5  —  c,  is  to  be  multiplied  by  that  represented 
by  c?  —  e ;  if  then  a  =  3,  6  =  2,  c=l,  o?  =  5,  ^  =  2,  a  +  S 
_  c  =  4,  c?  —  e  =  3,  and  .-.  (a  +  6  —  c)  .  (c?  —  e)  =  4  X  3 
=  12. 

Great  care  must  be  taken  in  observing  how  brackets  are 
employed,  and  what  effects  arise  from  the  use  of  them.  Thus 
(a  +  6)  .  (c  +  (i),  (a  +  6)  c  +  G?,  a  +  6c  +  cZ,  are  three  very 
different  expressions ;  for  if  a  =  3,  6  =  2,  c  =  3,  c?  =  5, 

(1.)  {a+b)  .  (c+cZ)  =  (3+2)  .  (3+5)=5x8=40. 

(2.)  (a+6)c+c?=(3+2)  3+5=5x3+5=20. 

(3.)  a+6c+cZ=3+2x3+5=3+6+5=14. 

17.  Instead  of  brackets,  a  line  called  a  vinculum  is  some- 
times used,  and  is  drawn  over  quantities,  which  are  taken  col- 
lectively.    Thus,  a—b—c  is  the  same  as  a---(6— c). 

The  line  which  separates  the  numerator  and  denominator 
of  a  fraction  may  be  regarded  as  a  sort  of  vinculum,  cor- 


16.  When  are  Jr«<;^6^5  employed  ? — 17.  What  is  a  vinculum?  May  tho 
line  which  separates  the  numerator  and  denominator  of  a  fraction  be  re- 
garded as  a  vinculum  ? 

1* 


6  ALGEBRA. 

responding,  in  fact,  in  Division  to  the  bracket  in  Multiplic(u 
tion.     Thus, —  implies  that  the  whole  quantity  a+5— c 

is  to  be  divided  by  5. 

18.  Like  quantities  are  such  k^  consist  of  the  same  letter^  or 
the  sa7ne  combination  of  letters ;  thus,  5a,  and  7«,  4a6  and 
9a6,  2bx^  and  Qhx^,  &;o.,  are  called  like  quantities ;  and  utu 
like  quantities  are  such  as  consist  of  different  Utters^  or  of  dif- 
ferent combinations  of  letters ;  thus,  4a,  36,  lax^  bbz^^  <Sjc., 
are  unlike  quantities. 

19.  Algebraic  quantities  have  also  different  denominations 
according  to  the  number  of  terms  (connected  by  the  sign  + 
or  — )  of  which  they  consist:  thus, 

a,  26,  8a:r,  &c.,  quantities  consisting  of  one  term,  are  called 
simple  quantities. 

a+x^  a  quantity  consisting  of  two  terms,  is  called  a 
binomial, 

bx-^-y—z^  a  quantity  consisting  of  three  terms,  is  called  a 
trinomiaL 


CHAPTEE    II. 

ON    THE    ADDITION,    SUBTRACTION,    MULTIPLICATION,    AND    DIVI- 
SION,   OF    ALGEBRAIC    QUANTITIES. 

ADDITION. 

20.  Addition  consists  in  collecting  quantities  that  are  like 
into  one  sum,  and  connecting  by  means  of  their  proper  signs 
those  that  are  unlike.  From  the  division  of  algebraic  quan- 
tities into  2^<^sitive  and  negative^  like  and  unlike^  there  arise 
three  cases  of  Addition. 

Case  I. 
To  add  like  quantities  with  like  signs. 

21.  In  this  case,  the  rule  is  "  To  add  the  coefficients  of  the 
several  quantities  together,  and  to  the  result  annex  the  com- 

18.  What  are  like  and  what  are  unlike  quantities  ? — 19.  What  is  a  simph 
quantity  ?  What  is  a  binomial  and  what  a  trinomial  ? — 20.  In  what  doea 
addition  of  algebra  consist  ?    Into  how  many  cases  is  it  divided  ? 


ADDITION.  7 

mon  sign,  and  the  common  letter  or  letters  ;"  for  it  is  evident 
from  the  common  principles  of  Arithmetic,  if  +2^,  4-3a,  and 
-f-5a,  be  added  together,  their  sum  must  be  4-10«;  and  if 
— -36\  —  45\  and  —Sb\  be  added  together,  their  sum  must 
be  -1561  . 


Ex.  1. 


Ex.2. 


Ex.3. 


2x+  3a  — 

46 

7x'-{-  Sx2/- 

56c 

4a'- 

3a'+   1 

Sx+  2a- 

56 

9^^+  2:ry— 

76c 

2a'— 

a^+17 

4:r+   8a- 

7b 

ll;r^-f  5.ry- 

46c 

5a'- 

2a^+  4 

9x-\-  4a— 

66 

a;'+  4a:y— 

6c 

Sa'- 

7a^+  3 

5^4-  7a-_ 

96 

x'-h    9X7/- 

26c 

a^- 

a^+10 

23:r+24a— 316  29^'+23^y— 196c  15a^— 14a^+35 


Ex.4, 

3;r»+4;c'^—  X- 
2x'+  x'—Sx 
7x'-\-2x''—2x 
Ax'+x''—  X 


Ex,  5. 

7a^-3a^6+2a6^-36' 

4a«-  a^6+  ab'-  6* 

a^-2a^6-f3a6^— 56» 

5a8~3(f&+4a6^-26« 


Ex.  6. 

2x^7/— Sx+  2 
4x^7/— 2x+  1 
3a:V— 5.r4-10 


-  In  these  Examples  it  may  be  observed  that  some  of  the 
quantities  have  7io  coefficient  In  this  case,  unity  or  I  t* 
always  understood.  Thus,  in  adding  up  the  first  column  of 
Ex.  2,  we  say,  1  +  1  +  11+9+7=29;  in  the  third,  2+1+4 
+7+5=19;  and  so  of  the  rest. 

Case  IL 

To  add  like  quantities  with  unlike  signs^ 

22.  Since  the  compound  quantity  a+6— c+cf— c,  &c,,  is 
positive  or  negative,  according  as  the  sum  of  the  positive 
terms  is  greater  or  less  than  the  sum  of  the  negative  ones,  the 
aggregate  or  sum  of  the  quantities  2a — 4a+7a — 3a  will  be 
+2«,  and  of  the  quantities  76*— 56*+26^— 86*  will  be  —46*; 
for  ill  the  former  case  the  excess  of  the  sum  of  the  positive 


22.  State  the  rule  in  the  Ist  ease. 


8 


ALGEBRA. 


terms  above  the  negative  ones  is  2a,  and  in  the  latter  4b\ 
Hence  this  general  rule  for  the  addition  of  like  quantities  with 
unlike  signs,  "Collect  the  coefficients  of  the  positive  terms  into 
one  sum,  and  also  of  the  negative  ;  subtract  the  lesser  of  these 
sums  from  the  greater ;  to  this  difference^  annex  the  sign  of 
the  greater  together  with  the  common  letter  or  letters,  and  the 
result  will  be  the  sum  required." 

If  the  aggregate  of  the  positive  terms  be  equal  to  that  of 
the  negative  ones,  then  this  difference  is  equal'  to  0 ;  and  con- 
sequently the  sum  of  the  quantities  will  be  equal  to  0,  as  in 
the  second  column  of  Ex.  2,  following. 


Ex.  1. 

E^^2. 

Ex.  3. 

4:X'-^X+     4 

— 7a5+35c--  xy 

~5^^+13a:« 

-2a;^+  X-  5  • 

— .  ah+2hc+4:xy 

.     -'2x'-  4x^ 

Zx''-^x+   1 

Sab—  hc+2xy 

7x'+     x' 

7a^+2a?-  4 

—^ah+Uc-Sxy 

9x^-Ux' 

-  rr^-4a;+13 

5ab  —  Sbc+  xy 

-J3a:^~2a;« 

lla;^-9^4-  9 


-2ab 


+Sxy 


—  4x^-'Qx'' 


Ex.  4. 

4ir^—  2x+Sy 

-  x^+  4:X—  y 

7x^-'     x-{-9y 

9x^+2lx-2y 


Ex.5. 

5a^-2a5+  b^ 

-a3_|_  ab-2b' 

4a'-Sab+  b' 

2a^+4:ab—4:b^ 


)L'' 


Ex.  6. 


^4xy+2xyS 

—   x^—   xy—1 

Sxy+4xy-5 

'-9xY—2xy-\-9 


Case  III. 

23.  There  now  only  remains  the  case  where  unlike  quan- 
tities are  to  be  added  together,  which  must^e  done  by  col- 
lecting them  together  into  one  line,  and  annexing  their  proper 
signs  ;  thus,  the  sum  of  3a?,— 2a,+5J,— 4y,  is  3;r— 2a+5S— 
Ay ;   except  when  like  and  unlike  quantities  are  mixed  to- 


23.  State  the  rule  in  the  2d  and  8d  cases. 


SIMPLE   EQUATIONS. 


9 


gether,  as  in  the  following  examples,  where  the  expressions 
may  be  simplified,  by  collecting  together  suclf  quantities  as 
will  coalesce  into  one  sum. 


Ex.  1. 

Sah  +    X  —  y  Collecting  together  like  quan- 

4c    —  2y  +  a;  titles,  and  beginning  with  3a6, 

6ab  —  3c  +  c?  we  have  Sab  -f  bab  =  Sab ;  +  x 

4y    +  o;^  —  2y  +  a;  =  +  2a; ;  —  y  —  2y  +  4y 

—  2y  =  — •  y;  4c  —  3c  =  +  c  ; 

besides  which  there  are  the  two 
quantities  +  d  and  +  x^,  which  do  not  coalesce  with  any  of 
the  others  ;  the  sum  recpiired  therefore  is  Sab  '\-,2x  —  y 
+  c  +  d-\'x\ 


8a6  +  2a;  —  y  +  c  +  (^  +  a;^ 


Ex.2. 

^x^—2xy+l  — 3y+4a;» 
4y  -\Sx^^y'^+xy—  x^ 
5a;^ — 2a;  +y  —15+  y' 
Zx'—xy  — 14+2y  +  12a;^— 2.r. 


Here  4a;'— a;^=3a;' 
— 2a;y+a;i/=— a;y 
+  1_.15=_14 

-3?/+4y+y=+2y 
+4a;^+3a;^+5a;^=  +  12a?* 

-2a;=-2a;. 


SIMPLB  EQUATIONS. 

24.  When  two  algebraic  quantities  are  connected  together 
by  the  sign  of  equality  (=),  the  expression  is  called  an 
equation.  Equations,  in  their  application  to  the  solution  of 
problems,  consist  of  quantities,  some  of  which  are  known  and 
others  unknown.  Thus  2a;+3=:a;+7  is  an  equation  in  which 
X  is  an  unknown  quaiitity^  and  its  value  is  such  a  number  as 
wil]  make  2^+3  and  x-\-l  equal  to  each  other.  The  number 
which  here  satisfies  the  equation,  or  is  the  value  of  a?,  is  man- 
ifestly 4,  since  2x4+3=11,  and  4+7=11.  The  value  of 
the  unknown  quantity,  which  has  in  this  example  been  found 
by  inspection,  is  usually  obtained  by  a  direct  calculation 
which  is  call'ed  the  solution  of  the  Equation. 


24.  What  is  an  equation?  What  is  meant  by  the  solution  of  an  equation*' 


10  ALGEBKA. 

25.  In  effecting  the  solution,  the  several  steps  of  the  pro- 
cess must  be  conducted  by  means  of  the  following  axioms, 
and  in  strict  accordance  with  them  : — 

(1.)  'Things  which  are  equal  to  the  same  thing  are  equal  to 
one  another.  ^ 

(2.)  If  equals  be  added  to  the  same  or  to  equals,  the  sums 
will  be  equal. 

(3.)  If  equals  be  subtracted  from  the  same  or  from  equals, 
the  remainders  will  be  equal. 

(4.)  If  equals  be  multiplied  by  the  same  or  by  equals,  the 
products  will  be  equal. 

(5.)  If  equals  be  divided  by  the  same  or  by  equals,  the 
quotients  will  be  equal. 

(6.)  If  equals  be  raised  to  the  same  power,  the  powers  will 
be  equal. 

(7.)  If  the  same  roots  of  equals  be  extracted,  the  roots 
will  be  equal. 

These  axioms,  exclusive  of  the  first,  may  be  generalized^ 
and  all  included  in  one  very  important  principle,  which  should 
in  every  investigation  in  which  equations  are  concerned  be 
carefully  borne  in  mind ;  viz.,  that  whatever  is  done  to  one  side 
of  an  equation  the  same  thing  must  be  done  to  the  other  side, 
in  order  to  keep  up  the  equality, 

26.  If  an  equation  contain  no  power  of  the  unknown  quan- 
tities higher  than  the  Jirst,  or  those  quantities  in  their  simplest 
form,  it  is  called  a  Simple  liquation. 


ON  THE  SOLUTION  OF  SIMPLE  EQUATIONS   CONTAINING  ONLY   ONE 
UNKNOWN    QUANTITY. 

27.  The  rules  which  are  absolutely  necessary  for  the  solu- 
tion of  simple  Equations,  containing  only  one  unknown  quan. 
tity,  may  be  reduced  to  four,  each  of  which  will  in  its  proper 
place  be  formally  enunciated  and  exemplified. 

25.  What  are  the  axioms  employed  in  the  solution  of  equations,  and 
state  the  general  principle  which  is  based  upon  them  ? — 26.  What  is  a  sim- 
ple equation  ? 


SIMPLE  EQUATIONS.  U 


Rule  I. 

"  If  the  unknown  quantity  has  a  cbefRcient,  then  its  value 
may  be  found  by  dividing  each  side  of  the  equation  by  that 
coefficient ;"  and  the  foui^ation  of  the  Eule  is,  that  "  If 
equals  be  divided  by  the  same,  the  quotients  arising  will  be 
equal." 

Ex.1.    Let  2:r=14;    then   dividing  both  sides    of   the 

2x     14  2x  14 

equation  by  2,  we  have   --=— ;   but    77=^,  and  77-= 7, 

-r^     ^     X  T  .  ,        ax     h-\-c     ,        ax 

Ex.2.   Let  ax=zb+ci   then  — = ;   but  — =a; ;  .'. « 

a        a    ^  a         ' 

b-i-c 
*"    a 
Ex.  3.  Let  x+2x+4x+ex=62. 
Collecting  the  terms,  13a; =52. 
Dividing  both  sides  of  the  equation  by  13, 
ar=4. 

Ex.  4.  Let  6x—4:X+Sx—x=S6, 
The  terms  being  added  as  in  Case  2d  of  Addition, 
4ar=36. 
Dividing  each  side  of  the  equation  by  4, 
x=9. 

Ex.  5.  10^=150.  Ans.  a?=15. 

Ex.  6.  Sx+4x+7x=zS4:.  Ans.  x—  6. 

Ex.  7.  8a;— 5a;-|-4a;— 2a;=25.  Ans.  a?=  5. 

Ex.  8.  \2x—Zx'-4:X—x=z2A.  Ans.  x=  6. 

28.  Arithmetical  questions  may  with  great  ease  Be  ex- 
hibited under  the  form  of  an  equation,  and  it  will  be  seen  hy 
the  subjoined  examples  in  what  relation  arithmetical  and  alge^ 
braic  operations  stand  to  each  other  :  as  for  instanccj 

If  3O5.  be  given  for  5 lbs.  of  tea^  what  is  the  price  of  1  lb.? 

(1.)  Tho,  price  of}  lb.  is  that  which  is  to  be  found.. 


12  ALGEBRA. 

(2.)  Then  it  is  clear  that  the  price  o/*l  lb.  X5  must  give  the 
price  of  5  lbs. 

(3.)  But  the  cost  of  5  lbs.  by  the  question=305. 

(4.)  Therefore,  the  price  of  1  lb.  in  shillings  X  5=305. 

(5.)  And  therefore  by  dividing  by  5,  we  obtain  the  price 
ofllb.=:65. 

The  several  steps  of  this  solution  expressed  algebraically 
would  take  the  following  more  compendious  form : 

(1.)  Let  ir=the  price  of  1  IJb.  in  shillings. 

(2.)  Then  5a;=the  price  of  1  lb.  in  shillings  X  5. 

(3.)  But  the  cost  of  5  lbs.  is  by  the  question=305. 

(4.)  .•.5a;=:305. 

(5.)  and  /.  x=Qs.  which  is  the  price  of  1  lb.,  as  was  re- 
quired. 

It  will  be  seen  by  steps  (2)  and  (3)  of  this  example,  that 
there  are  two  distinct  expressions  for  the  same  thing,  and 
that  in  step  (4)  these  expressions  are  made  equal  to  each 
other.  In  framing  equations  from  problems,  this  will  in  all 
cases  take  place.  As  a  second  example  let  this  problem  be 
taken: — 

A  house  and  an  orchard  are  let  for  £28  a  year,  but  the  rent 
of  the  house  is  6  times  that  of  the  orchard.  Find  the  rent  of 
each. 

The  rent  of  the  house  is  equal  to  that  of  G  orchards;  we 
may  therefore  change  the  house  into  6  orchards,  and  we  shall 
have 

Bent  of  the  orchard+6  times  rent  of  the  orchard=£2S, 
Taking  the  sum  of  the  rents  of  the  orchard,  we  get 

7  times  the  rent  of  the  orchard z=z £28, 
ftnd  the  7th  part  of  each  side  of  the  equation  being  taken, 

The  rent  of  the  orchard=£A  ; 
.and  .•.  the  rent  of  the  house=6  times  th6  rent  of  the  orchard 

=6  times  £4 
^  =£24.  _        .     .    • 

Now  to  give  to  these  operations  an  algebraical  shape, 
Let  x=:the  rent  of  the  orchard  in  £ 
then  6x=        "        "    house      " 


SIMPLE  EQUATIONS.  13 

But  by  the  condition  of  the  question, 

Rent  of  the  orchard-{-6  times  rent  of  the  orchard=£2S, 

or,       lfx=£2S. 
and,  dividing  each  side  of  t?^  equation  by  7, 
a: =£4,  the  rent  of  the  orchard, 
and  .-.  6x=6x£^=£'24,  the  rent  of  the  house. 

Again,  suppose  the  following  arithmetical  question  was 
proposed  for  solution ;  viz.,  "  To  divide  the  number  35  into 
two  such  parts,  that  one  part  shall  exceed  the  other  part 
by  9."  A  person  unacquainted  with  algebra  might  with 
no  great  difficulty  solve  this  question  m  the  following 
manner : — 

(1.)  It  appears,  in  the  first  place,  that  there  must  be  a 
greater  and  a  less  part. 

(2.)  The  greater  part  must  exceed  the  less  by  9. 

(3.)  But  it  is  evident  that  the  greater  and  less  parts  added 
together  must  be  equal  to  the  whole  number  35. 

(4.)  If  then  we  substitute  for  the  greater  part  its  equivalent, 
viz.,  "  the  less  part  increased  hy  9'',"  i^  follows,  that  the  less 
part  increased  by  9,  with  the  addition  of  the  said  less  part  is 
equal  to  35. 

(5.)  Or,  in  other  words,  that  twice  the  less  part  with  the 
addition  of  9,  is  equal  to  35. 

(6.)  Therefore,  twice  the  less  part  must  be  equal  to  35, 
with  9  subtracted  from  it. 

(7.)  Hence,  twice  the  less  part  is  equal  to  26. 

(8.)  From  whidh  we  conclude,  that  the  less  part  is  equal  to 
26  divided  hy  2;  i.  e.,  to  13. 

(9.)  And  consequently,  as  the  greater  part  exceeds  the  less 
Dy  9,  it  must  be  equal  to  22. 

But  by  adopting  the  method  of  sUgebraiG  notatio7i,  the  differ- 
ent steps  of  this  solution  may  be  much  more  briefly  expressed  as 
follows : — 

(1.)  Let  the  less  part      -    -    -    -         -     z=:x. 

(2.)  Then  the  greater  part ==x-\-9, 

(3.)  But  greater  part + less  part   -    -    -     =35. 
2 


14  ALGEBRA. 

(4.)  .-.  x-h9+x     ...  ...  =35. 

(5.)  or  2a;+9   - =35. 

(6.)  .\2x =35-9. 

(7.)  or  2a; -     -    -  =26. 

^  26 

(8.)  ,' .  X  (less -psLTt) =—=13. 

{9.)  SLXid  x+9  {greaier  ^SLTt)    ....     =13+9=22. 

29.  Having  thus  explained  the  manner  in  which  the  several 
steps  in  the  solution  of  an  arithmetical  question  may  be  ex- 
pressed in  the  language  of  Algebra,  we  now  oroceed  to  its  ex- 
emplification. 

PROBLEMS. 

Prob.  1.  A  dessert  basket  contains  30  apples  and  pears, 
but  4  times  as  many  pears  as  apples.  How  many  are  there 
of  each  sort  ? 

Let  a;=the  number  of  apples; 
then,  as  there  are  4  times  as  many  pears  as  apples, 
4x= the  number  of  pears. 
But  by  the  question  the  apples  and  pears  together =30. 

.-.  a;+4;r=30. 
Adding  the  terms  contaming  x, 

5a; =30. 
Dividing  each  side  of  this  equation  by  5, 

ir=6,  the  number  of  apples, 
/.  the  number  of  pears =4a: =4x6 =24. 

Prob.  2.  In  a  mixture  of  16  lbs.  of  bla«k  and  green  tea, 
there  was  3  times  as  much  black  as  green.  Find  the  quantity 
of  each  sort. 

Let  a;=the  number  of  lbs.  of  green  tea, 
then3a;=  "         "  "  black. 

But  the  black  tea -f- the  green  tea=16  lbs. 

/.a;+3a;  =  161bs. 
Collecting  the  terms  which  contain  .r, 
4a;=161bs. 
Dividing  each  side  of  this  equation  by  4, 

^  a? =4  lbs.  of  green  tea, 

.-.  the  black  tea -3ar=3x4=121bs. 


SIMPLE  EQUATIONS.  15 

Prob.  3.  An  equal  mixture  of  black  tea  at  5  shillings  a  lb. 
and  of  green  at  7  shillings  a  lb.  costs  4  guineas.  How  many 
lbs.  were  there  of  each  sort  ? 

Let     a;=the  number  of  lbs.  of  each  sort; 
then  5x=zthe  cost  of  tke  black  in  shillings, 
and  7x=z         "         "      green         " 
But  cost  of  black + the  cost  of  green =4  guineas =845. 

/.  bx-\-l[X=zS4: 

12x=zS4 

.*.  x=7  lbs.  of  each  sort. 

Prob.  4.  The  area-of  the  rectangular  floor  of  a  school-room 
is  180  square  yards,  and  its  breadth  9  yards.  What  is  its 
length  ? 

Let  a: = the  length  in  yards;   then  since  the  area  is  the 
length  multiplied  by  the  breadth,  we  have 

irX9=:the  area  of  the  floor, 
.\9a;=:180 
and  .'.    a; =20  yards,  the  length  required. 

Prob.  5»  Divide  a  rod  15  feet  long  into  two  parts,  so  that 
the  one  part  may  be  4  times  the  length  of  the  other. 

Let  rr=the  less  part,  i i  ^^^- i 

then  4^= the  greater  part,  x  4x 

Now  these  two  parts  make  together  15  ft. 
.\x-^4:Xz=16fL 
5;r  =  15ft. 
/,x=z  3  ft.  the  less  part, 
and  the  greater  part =4  times  3  ft.  =:  12  ft. 

Prob.  6.  A  horse  and  a  saddle  were  bought  for  £40,  but 
the  horse  cost  9  times  as  much  as  the  saddle.  What  was  the 
price  of  each  ?  Ans.  £36  and  £4. 

Prob.  7.  Divide  two  dozen  marbles  between  Richard  and 
Andrew,  so  that  Richard  may  have  three  times  as  many  as 
Andrew  1  Ans,  Richard's  share  =  18. 

Ans,  Andrew's  share  =  6. 

Prob.  8.  A  boy  being  asked  how  many  marbles  he  had, 
-   f  ^fl.idj  If  I  had  twice  as  many  more,  I  should  have  36^    How 
many  had  he  1  Ans.  12. 

Prob.  9.  A  bookseller  sold  10  books  at  a  certain  price,  and 


16  '    ALGEBRA. 

afterwards  15  more  at  the  same  rate,-  and  at  the  latter  time 
received  355.  more  than  at  the  former :  what  was  the  price 
per  book  ? 

Let  rr=the  price  of  a  book  in  shillings  ; 
then  10a:=:the  price  of  thp  1st  lot  in  shillings, 
and   I5x=         "  "  ^  2d     " 

Now,  if  the  price  of  the  1st  lot  be  taken  from  that  of  the 
2d,  there  remains  a  difference  in  price  of  355. 
,\15x—l0xz=S5s,  ^ 

Subtracting  the  10a;  from  the  15:r,  we  have         ' 
5a; =355. 

.-.  x=7s.  the  price  of  a  book. 
•  Prob.  10.  Divide  £300  amongst  A,  B,  and  C,  so  that  A 
may  receive  twice  as  much  as  B,  and  C  as  much  as  A  and  B 
together. 

Let  a;=B's  share  in  £ 
then  2a;=z:A's  share  in  £ 
and  x-\-2x  or  3j;=rC's  share  in  £. 
But  amongst  them  they  receive  £300 ; 
.\  x-{-2x-\-Sx=£S00, 
6a:=£300  ; 
.-.  a; =£50  B's  share ; 
.-.  A's  shaT-e=£100,  and  C's  share=£150. 
Prob.  11.  If  to  nine  times  a  certain  number,  three  times 
the  numl^er  be  added,  and  four  times  the  same  number  be 
taken  away,  there  will  then  be   obtained  the  number  48. 
What  is  the  number  ? 

Let  a; = the  number; 
then  9a;=nine  times  the  number, 

3a: = three  times  the  number, 
and  4a; = four  times  the  number; 

.-.  9a;+3a;—4a;=48, 
'    •  .    8a;=48; 

.'.   x=6,  the  number  required. 

Prob.  12.  The  sum  of  £100  is  to  be  divided  among  2  men, 
3  women,  and  4  boys,  so  that  each  man  shall  have  twice  as 
much  as  each  woman,  and  each  woman  three  times  as  much 
as  each  boy.     Find  the  share  of  each. 
•      Let  each  boy's  shares  a;; 
then  each  woman's  share  =r 3a;, 
and  each  man's  share =2  times  3a;=:6a;; 


SIMPLE   EQUATIONS.  •         17 

Hence  we  have, 

the  share  of  the  4  boys     =4rr, 
the  share  of  the  3  women =3  times  Sxz=z9x, 
and  the  share  of  the  2  menr=:2  tunes  6^  =  12^ : 
But  the  sum  of  all  these  shares  is  to  amount  to  £100 ; 
/.4.^+9a;+^2a;==£100, 
25:r=£100 ; 

,\  each  boy's  share =£4 ;   each  woman's =3  times  £4 
p  ^  A  =£12  ;  and  each  man's=6  times  £4=:£24. 

Prob.  13.  a  gentleman  meeting  4  poor  persons  gave  five 
shillings  amongst  them  ;  to  the  second  he  gave  twice,  to  the 
third  thrice,  and  to  the  fourth  four  times  as  much  as  to  the 
first.     What  did  he  give  to  each  ] 

Ans,  6d.,  12c/.,  18d,  24c?.,  respectively. 

Prob.  14.  Divide  a  line  12  ft.  long  into  three  parts,  such 
that  the  middle  one  shall  be  double  the  least,  and  the  greatest  y/L 
triple  the  least  part.  Ans,  2,  4,  6. 

Prob.  15.  Divide  40  into  three  such  parts,  that  the  first 
shall  be  5  times  the  second,  and  the  third  equal  to  the  differ- 
enee  between  the  first  and  second.  A7is.  20,  4,  16. 

Prob.  16.  A  grocer  mixed  three  kinds  of  tea,  Bohea  at  35. 
per  lb.,  Twankay  at  5^.,  and  Souchong  at  7^.  per  lb.  The 
mixture  contains  the  same  quantity  of  each,  and  Cost  £6. 
How  many  lbs.  are  there  of  each  kind  1  Ans.  8  lbs. 

Prob.  17.  A  bill  of  £700  was  paid  in  sovereigns,  half- 
sovereigns,  and  crowns,  and  an  equal  number  of  each  was  used. 
Find  the  number.  Ans,  400. 

Prob.  18.  Two  travellers  set  out  at  the  same  time  from 
Guildford  and  London,  a  distance  of  27  miles  apart ;  the  one 
walks  4  miles  an  hour,  and  the  other  5  miles.  In  how  many 
hours  will  they  meet  ?  Ans.  S  hours. 

Prob.  19.  A  person  bought  a  horse,  chaise,  and  harness, 
for  £120  ;  the  price  of  the  horse  was  twice  the  price  of  J;he 
harness,  and  the  price  of  the  chaise  twice  the  price  of  both 
horse  and  harness  ;  what  was  the  price  of  each  ? 

(  Price  of  harness =£13     6     8 
Answer,  \      "      "  horse    z=   26  13     4 
(      "     "  chaise  =   80     0     0 
2* 


18  ALGEBRA. 


SUBTRACTION. 


30.  Subtraction  is  the  finding  the  difference  between  two 
algebraic  quantities,  and  the  connecting  them  by  proper  signs, 
so  as  to  form  one  expression^*  thus,  if  it  were  required  to 
subtract  5—2  (^^  e.,  3)  from  9,  it  is  evident  that  the  remainder 
would  be  greater  by  2  than  if  5  only  were  subtracted.  For 
the  same  reason,  if  6— c  were  subtracted  from  a,  the  remain- 
der would  be  greater  by  c,  than  if  h  only  were  subtracted. 
Now,  if  b  is  subtracted  from  a,  the  remainder  is  a — 5  ;  and 
consequently,  if  5— c  be  subtracted  from  a,  the  remainder 
will  be  a—b-\-c.  Hence  this  general  Rule  for  the  subtraction 
of  algebraic  quantities,  "  Change  the  signs  of  the  quantities  to 
he  subtracted^  and  then  place  them  one  after  another,  as  in 
Addition." 

Ex.  1.  From  5cH-3a;— 25  take  2c --4y.  The  quantity  to 
be  subtracted  w;i7A  its  signs  changed,  is-— 2c -{-Ag  ;  therefore 
the  remainder  is  5a+3a;— 25— 2c+4y. 

Ex.2.    From   7x^-^2x +5  tsike  Sx'^+^x-^l -, 
The  remainder  is  7x^ — 2a;  +  5    —Sx^—5x-{-l] 

or  7x^-2x'^2x  -^x  +5   +1=4^;'^- 7a; +6. 

But  when  like  quantities  are  to  be  subtracted  from  each 
other,  as  in  Ex.  2,  the  better  way  is  to  set  one  row  under  the 
other,  and  apply  the  following  Rule  ;  "  Conceive  the  signs  of 
the  quantities  to  be  subtracted  to  be  changed,  and  then  proceed 
as  in  Addition." 

Ex.  3.  Ex.  4.       I  Ex.  5.    , 

YTom7x'-2x+5  l2a^-~Sa+  b-l  5y^-4y+3a 

Subtract  Sx'+6x-^l  6a'-\-  a-2b+S  6/-4y-  a 

Remainder  4:x^—lix-{-6  6a^— 4a+35— 4  —y^     *  +4a 

*Ex.  6.  Ex.  7.  Ex.  8. 

From  7xg+2x-'Sg       Ux+g—z—  5       lSx^-2x''^7 
Subtract  3a:?/—  x-\-  y  x-^y+z—\\       — aj^-f-  x^—Q 


80.  What  is  suUraction  ?    State  the  rule  for  the  subtraction  of  algebraie 
quantities,  and  explain  the  principle  on  which  it  rests. 


SIMPLE   EQUATIONS.  19 


OP  THE  SOLUTION  OF  SIMPLE  EQUATIONS,  CONTAININ(i  ONLY  ONE 
UNKNOWN  QUANTITY. 

]^LE    11. 

81.  "Any  quantity  may  be  transferred  from  one  side  of 
the  equation  to  the  other,  by  changing  its  sign ;"  and  it  is 
founded  upon  the  axiom,  that  "If  equals  be  added  to  or 
subtracted  from  equals,  the  sums  or  remainders  will  be 
equal." 

Ex.  1.  Let  a;+8  =  15;  subtract  8  from  each  side  of  the 
equation,  and  it  becomes  ir-[-8— 8  =  15— 8;  but  8— 8=0, 
.•.a:=15-8=7. 

Ex.  2.  Let  a:— 7=20;  add  7  to  each  side  of  the  equation, 
thena:-7+7=20+7;  but -7+7=0;  .-.  a:=20+7=27. 

Ex.3.  Let  3a;— 5=2a;+9;  add  5  to  each  side  of  the 
equation,  and  it  becomes  3a;— 5  +  5=2a;+9  +  5,  or  3a;=2a;+ 
9+5.  Subtract  2x  from  each  side  of  this  latter  equation, 
then  Sx—2x=:2x—2x-{-9-^5',  but  2x—2x=z0,  .'.Sx—2x=:9 
+  5.     Now  3a;— 2a:=a;,  and  9  +  5  =  14;  hence  a;  =  14. 

On  reviewing  the  steps  of  these  examples,  it  appears 

(1.)  Thatir+8=15  is  identical  with  a;=15— 8. 

(2.)     "     a;-7=20  "         with  a;=20+7. 

(3.)     "    3a;-5=2a;+9       "         with  3a; -2a; =9 +5. 

Or,  that  "  the  equality  of  the  quantities  on  each  side  of  the 
equation,  is  not  affected  by  removing  a  quantity  from  one  side 
of  the  equation  to  the  other  and  changing  its  sign;"* 

From  this  rule  also  it  appears,  if  the  same  quantity  with 
the  same  sign  be  found  on'hoth  sides  of  an  equation,  it  may 
be  left  out  of  the  equation;  thus,  ifa;+a=c+a,  then  a;=c+ 
a — a  ;  but  a — a=0,  .*.  a;=c. 

It  further  appears,  that  the  signs  of  all  the  terms  of  an  equa- 
tion may  be  changed  from  +  to  — ,  or  from  —  to  +,  without 
altering  the  value  of  the  unknown  quantity.  For  let  a;— 6=c 
— ^  •  then  bv  the  Rule.  a;=c— a+6;  chano-e  the  si"-ns  o^  all 


20  ALGEBRA. 

the  terms,  then  5—  irs=:a— c,  in  which  case  b—a-{-c=x,  or  x^=. 
c— a+6,  as  before. 

Ex.4.       2a;+3=a;+n.  Ans,  x=U, 

Ex.  5.       5x—4:z=4tx+26.  Ans,  x=29. 

Ex.  6.       7^— 9=r6a;— 3.        C  Ans,  x^(S, 

Ex.  7.       4:X+2a=Zx+^h,  Ans.  ir=96— 2a. 

Ex.8.     15:r+4=34.  Ans.  x=2. 

Ex.  9.       8^+7=6^+27.  -4^5.  a;=10. 

Ex.  10.     9^—3=4^+22.  Ans.  x—b. 

Ex.  11.  17a;— 4a;+9=:3ar+39.  Ans.  x=^. 

Ex.  12.     ax—cz=zh+2c,  Ans.  x= . 

a 

Ex.  13.     5;r-(4a;— 6)  =  12. 

The  sign  —  before  a  bracket  being  the  sign  of  the  whole 
quantity  enclosed,  indicates  that  the  quantity  is  to  he  sub- 
tracted;  and  therefore,  according  to  the  Rule,  when  the  brack- 
ets are  removed  the  sign  of  each  term  must  be  changed. 
Thus,  the  signs  of  4a;  and  of  6  are  respectively  +  and  — ,  but 
when  the  brackets  are  removed  they  must  be  changed  to  — 
and  +  respectively.  The  equation  then  becomes 
5a;-4a;+6  =  12. 
By  transposition,  5a; — 4a; = 1 2 — 6  ; 

.•.a;=6. 
Ex.  14.  6a;-(8+a;)=4a;-(a?-10). 

By  removing  the  brackets,  and  changing  the  signs  of  the 
terms  w^hich  they  enclose,  the  equation  becomes 
6a;— 8— a;=4=4.r— OJ+IO. 
Transposing,  6a; — a; — 4a; + a; = 1 0 + 8  ; 

.•.2a;=18. 
Dividing  both  sides  of  the  equation  by  2, 

a;=9. 
Ex.15.     4a;-(3a;+4)=8.  Ans.  x=z\2. 

Ex.  16.     8a;— (6a;— 8)=9— (3— a;).  ^^5.  a;  =—2. 

e    Ex.  17.     4a;-(3a;-6)-(4a;-12)  =  12-(5a;-lG). 

Ans.  x=2. 
Ex.  18.     5a;— (3-4-aai)=8  — (— a;— 1).        Ans.  a;=12.  . 


SIMPLE   EQUATIONS.  21 


PROBLEMS. 


Prob.  1.  There  are  two  numbers  whose  difference  is  15  and 
their  sum  59.     What  are  the  numbers  ? 

As  their  difference  is  15,  itjs  evident  that  the  greater  num- 
ber must  exceed  the  less  by  15. 

Let  therefore  ir=: the  less  number; 
then  will  a; +15=:  the  greater  : 
But  their  sum=:59 ; 
.-.  a;+a;+15=59, 
or2a;+15=59. 
And,   transposing  15,  2a; =59 —  15, 
or  2a; =44; 
.•.  a:=22  the  less  number, 
and  a;+15=22+15=37  the  greater. 

Prob.  2.  I  gave  to  Richard  and  James  27  marbles,  l&ut  to 
Eichard  5  more  than  to  James.  How  many  did  I  give  to 
each? 

Let  ir=the  number  I  gave  to  James ; 
thena;+5=  "         "  "      "  Richard: 

But  together  they  receive  27  ; 
.•.a;+a;+5=27, 
or2a;+5=27. 
Transposing,  2a; =27— 5, 

or  2a; =22; 
.*.  a;=ll,  the  No.  James  received, 
anda;+5=16     "     "     Richard  received. 

Prob.  3.  Four  times  a  number  is  equal  to  double  the  num- 
ber increased  by  12.     What  is  the  number? 
Let  a;=the  number ; 
then  4a; =4  times  the  number, 
2a; = double  tlie  number, 
and  2a;+12=double  the  number  increased  hy  12. 
Therefore,  by  the  equality  stated  in  the  question, 

4a;=2a;+12. 

By  transposition,  4a;— 2a; =12. 

2a;  =  12; 

,\xt=l   6. 

Prob.  4.  At  an  election  420  persons  voted,''and  the  success- 


22  ALGEBRA. 

ful  candidate  had  a  majoi^ity  of  46.     How  many  voted  for 
each  candidate  ?  Aris,  187  and  233. 

Prob.  5.  One  of  two  rods  is  8  feet  longer  than  the  other, 
but  the  longer  rod  is  three  times  the  length  of  the  shorter. 
What  are  their  lengths  ?  |  Ans.  4  ft.  and  12  ft. 

Prob.  6.  Five  times  a  number  diminished  by  16  Is  equal 
to  three  times  the  number.    What  is  the  number  ?    Ans.  8. 

Prob.  7.  A  horse,  a  cow,  and  a  sheep,  were  bought  for 
£24;  the  cow  cost  £4  more  than  the  sheep,  and  the  horse  £10 
more  than  the  cow.     What  was  the  price  of  the  sheep  1 
Let  a:=the  price  of  the  sheep  in  £ ; 
then  a;+4=  "         "      cow         " 

and  a;+4-f  10=  "         "      horse      " 

But  these  three  prices  taken  together  amount  to  £24 ; 

...  rrH-(a;+4)  +  (a;+4+10)=24. 
Ad^ng  together  like  terms, 

3a;+18=z24. 
By  transposition,  3;r = 24 — 1 8, 
Sx=6; 
.'.  x=z£2,  the  price  of  the  sheep. 
Prob.  8.  A  draper  has  three  pieces  of  cloth,  which  togethei 
measured  159  yards;  the  second  piece  was  15  yards  longei 
than  the  first,  and  the  third  24  yards  longer  than  the  second. 
What  is  the  length  of  each  piece  ? 

Ans.  35,  50,  and  74  yds. 
Prob.  9.  Divide  £36  among  three  persons.  A,  B,  and  C,  in 
such  a  manner  that  B  shall  have  £4  more  than  A,  and  C  £7 
more  than  B.  Ans.  £7,  £11,  and  £18. 

Prob.  10.  A  gentleman  buys  4  horses  ;  for  the  second  he 
gives  £12  more  than  for  the  first ;  for  the  third  £5  more  than 
for  the  second ;  and  for  the  fourth  £2  more  than  for  the  third. 
The  sum  paid  for  all  the  horses  Was  £240.  Find  the  price  of 
each.  Ans.  £48,  £60,  £65,  and  £67. 

Prob.  11.  What  number  is  that  whose  double  is  as  much 
above  21  as  it  ie  itself  less  than  21  ? 
Let  ir  =  the  number ; 
then  2.r=:  double  the  immber, 
2:r— 21i=:what  double  the  number  is  above  21, 
and  21— a: = what  the  number  is  less  than  21 : 


SIMPLE   EQUATIONS.  23 

But  bj  the  question  these  two  values  are  equal  to  each 
other  ;  /.  2^—21  r=21  — ar. 

By  transposition,  2a;+ic=21+21, 
3a: =42; 
.-..^  =  14. 
The  answer  may  easily  be%)roved  to  be  correct,  for  2a;— 21 
=28-21=7,  and  21-a:rz:^l-14=7  ;   that  is,  twice  14  is 
as  much  above  21,  as  21  is  above  14,  namely  7. 

Prob.  12.  In  dividing  a  lot  of  oranges  among  a  certain 
number  of  boys  I  found  that  by  giving  4  to  each  boy  I  had 
6  to  spare,  and  by  giving  3  to  each  boy  I  had  12  remaining. 
How  many  boys  were  there  ? 

Let  ir=:the  number  of  boys  ; 
then,  if  I  gave  to  each  boy  4  oranges,  I  should  give  away  4 
times  X  oranges ; 

.*.  4:r=number  of  oranges  distributed  at  first ; 
But  the  total  number  of  oranges  is  6  more  than  this  num. 
ber; 

/,  Total  number  oi  OT2iT\gQs-=4,x-\-^: 
Again,  if  each  boy   received   3   oranges,  there  were   12 
oranges  left ; 

.'.  Total  number  of  0V2ingQS=^x-\-\2, 
These  two  values  for  the  number  of  oranges  expressed  in 
terms  of  a;  must  necessarily  be  equal ;  Axiom  (1.)      / 

.-.4^+6=3^+12.  MJ/^t^) 

By  transposition,  4a; — 3a: = 1 2 — 6 ;  '    ^""^^ 

.*.  a; =6,  the  number  of  boys. 

Prob.  13.  An  express  set  out  to  travel  240  miles  in  4  days, 
but  in  consequence  of  the  badness  of  the  roads,  he  found  that 
he  must  go  5  miles  the  second  day,  9  the  third,  and  14  the 
fourth  day  less  than  the  first.  How  many  miles  must  he 
travel  each  day  ? 

Let  a;=:the  number  of  miles  on  the  1st  day  ; 
thena:-5=  "  "  "      2d     " 

a:— 9=  "  "  "      3d     " 

and  a: -14=  «  "  "      4th   " 

Now  the   number  of  miles  which  he  travels  in 
=240. 

...  a;+a;—5  +  a:—9+a:  — 14=240. 
Collecting  the  terms,  4a:— 28=240. 


24  ALGEBRA. 

By  transposition,         4x = 240  -h  28, 
4.r=268; 
.*.  re =67,  the  number  of  miles  he  goes  on  1st  day, 
a:-5  =  62    "  "  "         "         "       2d     " 

a:-9=58    "  "  "         "         "       3d     " 

and  :^-14r=:53    "         "  i         "        *      4th  " 

pROB.  14.  It  is  required  io  divide  the  number  99  into  five 
such  parts  that  the  first  may  exceed  the  second  by  3,  be  less 
than  the  third  by  10,  greater  than  the  fourth  by  9,  and  less 
than  the  fifth  by  16. 

Ans.  The  parts  are  17,  14,  27,  8,  33. 

Prob.  15.  Tv70  merchants  entered  into  a  speculation,  by 
which  A  gained  £54  more  than  B.  The  whole  gain  was  £49 
less  than  three  times  the  gain  of  B.  What  were  the  gains  ?  ' 
A71S.  A's  gain=r:£157  ;  B's=£103 


Prob.  16.  In  dividing  a  lot  of  apples  among  a  certain  num- 
ber of  boys,  I  ^und  that  by  giving  6  to  each  I  should  have  too 
few  by  8,  but  by  giving  4  to  each  boy  I  should  have  12  re- 
maining.    How  many  boys  were  there?  Ans,  10. 


MULTIPLICATION. 

32.  Multiplication  is  the  finding  the  product  of  two  or 
more  algebraic  quantities ;  and  in  performing  the  process,  the 
four  following  rules  must  be  observed. 

(1.)  When  quantities  having  like  signs  are  multiplied  to- 
gether, the  sign  of  the  product  will  be  +  ;  and  if  their  signs 
are  unlike,  the  sign  of  the  product  will  be  —  .* 

*  This  rule  for  the  multiplication '"of  the  Signs  may  be  thus  ex- 
plained : — 

I.  If  +a  is  to  be  multiplied  by  -|-6,  it  means,  that  -fa  is  to  be  added 
to  itself  as  often  as  there  are  units  in  b,  and  consequently  the  product 
will  be  -f  <*^- 

II.  If  — a  is  to  be  multiplied  by  -\-b,  it  means,  that  — a  is  to  be 
added  to  itself  as  often  as  there  are  units  in  b,  and  therefore  the  product 
is  — ab. 

82.  What  is  multiplication,  and  what  are  the  Kules  to  be  observed  m 
multiplication  ? 


MULTIPLICATION.  25 

(2.)  The  ooefficients  of  ikiQ  factors  must  be  multiplied  to- 
gether, to  form  the  coefficient  of  the  product, 

(3.)  The  letters  of  which  they  are  composed  must  be  set 
l-'>wn,  one  after  another,  according  to  their  order  in  the 
Alphabet,  ^ 

(4.)  If  the  same  letter  is  found  in  both  factors,  the  indict 
of  it  must  be  added  together,  to  form  the  index  of  it  in  the 
product. 

Thus,  +a  multiplied  by  +6  is  equal  to  +ab^  and  —a  mul- 
tiplied by  —b  is  also  equal  to  -\-abi  +3a;X —%  =  — 15a;y ; 
-SabX+4:cd=-12abcd  ;  -4:a'b^X  -Sabd'=-hl2a'b'd*  ; 
&c.,  &c. 

From  the  division  of  algebraic  quantities  into  simple  and 
compound,  there  arise  three  cases  of  Multiplication.  In  per- 
forming the  operation,  the  Rule  is,  "To  multiply  Jlrst  the 
signs,  then  the  coefficients,  and  afterwards  the  letters." 


Case  I. 

33.  When  both  factors  are  simple  quantities ;  for  which  the 
Eule  has  been  already  given. 

III.  If  +  a  is  to  be  multiplied  by  —  6,  it  meana,  that  +«  is  to  be 
subtracted  as  often  as  there  are  units  in  b,  and  consequently  the  product 
is  — ab. 

IV.  If  —  a  is  to  be  multiplied  by  —  b,  it  means,  that  —  a  is  to  be 
subtracted  as  often  as  there  are  units  in  b ;  and,  since  to  subtract  <i 
negative  quantity  is  the  same  as  to  add  a  positive  one,  the  product  will 
be  -f-  <^b. 

Or,  these  four  Rules  might  be  all  comprehended  in  one  ;  thus, 

To  multiply  a  —  6  by  c  —  c?,  is  to  add  a  —  6  to  itself  as  often  as  there 
are  units  in  c  —  d]  now  this  is  done  by  adding  it  c  times,  and  subtract- 
ing it  d  times ; 

But  a  —  b,  added  c  times  .  .  .  =ac  —  be, 
and  a  —  b,  subtracted  d  times    =  —  «c?  -f-  bd, 

,'.  a  —  b  X  c  —  d =ac  —  be  —  ad-i-bd. 

i.  e.,  -{-  a  X  -{-  c  =:^ -\-  ac 
—  b  X  +c  =  —  bc 
-{-aX  —  d  =  —  ad 
^bX  —  d  =  -\-bd, 

3 


26 


ALGEBRA. 


Ex.  1. 

4ab 
Sa 


Ex.  2. 

2axy 
~3y 


Ex.  a. 

— Sahc 
ba% 


Ex.  4. 

—  ^a^bc 

-  2h''x' 


12a^b 

—6axy' 

Ex.6. 

~2y 

-^l^a'b'c 

+  10a%'cx^ 

Ex.5. 

4abc 
Sac 

Ex.  7. 

— 4cdx 

2c 

Ex.8. 
'-2ac^x 

Case  II. 

84.  When  one  factor  is  compound  and  the  other  simple, 
"  Then  each  term  of  the  compound  factor  must  be  multiplied 
by  the  simple  factor  as  in  the  last  Case,  and  the  result  will 
be  the  product  required." 


Ex.1. 

Multiply  Sab—2ac-\-d 
by  4a 

Ex.2. 

Sx^      2x^+A 
— 14aa; 

Product  \2a%^Sa''c+^ad 

—A2ax^-\-2Sax^—bQa^ 

Ex.  3. 

Multiply         Ix""  —2a:  +4a 
by  —  Sa 

Ex.4. 

12a3-2a2+4a-l 
Sx 

Product    — 21aa;^  +  6aa;— 12a^ 

Ex.  5. 

Multiply      ^a'^x+Sa—x+l 
by  —  x^ 

Product 

Ex.  6. 

Ax^y+Sx—2y 
Sxy 

Case  IH. 

35.  When  both  factors  are  compound  quantities,  each  term 
of  the  multiplicand  naust  be  multiplied  by  each  term  of  the 


\ 


MTJXTIPLICATION.  27 

multiplier ;   and  then  placing  like  quantities  under  each  othet 
the  sum  of  all  the  terms  will  be  the  product  required. 

Ex.  1.  Ex.  2.  Ex.  3. 

Multiply  a  +     b  (P+  b  a^+ab+b^ 

hy  a  -{•     b  a  —  b  a  —  b 


1st,  by  a      a^+  ab  a^+ab  a^+a^b-{-ab^ 

2d,'byi  ab-hP  -ab^b''  ^a^h-ab''-^b^ 

Product  a'^+2a^>+62       a^     *  -^^         "Z~^       *~~^» 


Ex.  4.  .  Ex.  5. 

3^*+  ^x  3a:2-  2a; +5 

4.^+7  6a;  -  7 


12a:^+  8a;«  18a;^-12a;^+30a: 
4-21a;^+14a;  ~21a;'+14a;--35 

12^^+29^+14^  18a:«-33a:2+44a:-35 


Ex.  6. 

14a  c  —  3a  5  +  2 
ac  —     a  6  +   1 

14a V-  3a^6c+~2^ 

-14a^6c  +3a'»62-2aJ 

+14ac     — 3a6+2 

14aV-17a'^6c+16ac+3a^62-5a6+2 


Ex.  7. 

Ex.  8.  Multiply  a3+3a^6+3a5^+62      by      a+ft. 

^n5.  a*+4a^5+6a262+4a6M  i^*- 


28  ALGEBRA. 

Ex.  9.  Multiply  4a:V+3^y— 1   -  -  by  ^x^—x, 

Ans,  ^x^'y+^x^y-^^x^-'Zx^y+x. 

Ex.  10.       "        (x^-^x^+x-^ by  ^x'^+x+l. 

Arts.  2^'— icHS.'i;^— lOo;^— 4a;— 5. 

Ex.  11.       «         3a«+2a6-52 ^^  ^a?^'^ah+h\ 

Ans.  9a*-4a262+4a63-6^ 

Ex.  12.      "        oi^+x^y+xy^+y^     by  x^y, 

Ans,  a;*— 2/\ 

Ex.  13.       «        x^-'lx+l  -  -  -  .    by  x'^^lx. 

/Ans.  x^—\x^+^-^x^--\x. 

OH  THE   SOLUTION  OP  SIMPLE  EQUATIONS  CONTAINING  ONLY  ONE 
UNKNOWN   QUANTITY. 

Ex.  1.     3a;+4(a;+2)=36. 

The  term  4  (a; +2)  means,  that  a: +2  is  to  be  multiplied  by 
4,  and  the  product  by  Case  2d  is  4a;+8 ; 
.-.  3a:+4a?+8=36. 
Adding  together  the  terms  containing  ar,  and,transposing  8, 
7ir=36-8, 
7a;=:28; 
/,  ii;=4. 

Ex.2.     8(a?+5)+4(a;+l)=8a 
Performing  the  multiplication, 

8a;+40+4a;+4=80. 
Collecting  the  terms,  12a; +44 =80. 
Transposing,  1 2a; = 80 — 44, 

12a;=36; 

.-.  a?=3. 
Ex.3.     6(a;+3)+4a;=58.  Ans,  x=^4. 

Ex.4.  30(a;-3)+6=6(a:+2).  Ans.  a;=4. 

Ex.5.     5(a;+4)~3(a;-5)=49.  Ans.  x:=l 

Ex.6.     4(3+2a;)-2(6-2a;)=60.  Ans.  x-^b 

Ex.7.     3(a;-2)+4=4(3-a:).  Ans.  x==^2. 

Ex.8.     6(4-~.a;)— 4(6— 2a;)-12=0.  Ans.  X'-=.Q. 


SIMPLE   EQUATIONS.  29 


PROBLEMS. 

Prob.  1.  What  two  numbers  are  those  whose  difference  is 
9,  and  if  3  times  the  greater  be  added  to  5  times  the  less,  the 
•eum  shall  be  35  ?  -I 

Let  ar=the  less  number; 
then  a; +9= the  greater. 
And  3  times  the  greater =8   (a; +9)= 3a; +27, 

5  times  the  less       =bx. 
But  by  the  problem,  3  times  the  greater  +  5  times  the  less 
=35; 

/.  Sx+27+5x=S5, 
8a;+27=35. 
Transposing,  8a;=35— 27=8; 

.'.  aj=l,  the  less  number, 
and  a;+9=10,  the  greater. 

Prob.  2.  A  courier  travels  7  miles  an  hour,  and  had  been 
dispatched  5  hours,  when  a  second  is  sent  to  overtake  him, 
and  in  order  to  do  this,  he  is  obliged  to  travel  12  miles  an 
hour.     In  how  many  hours  does  he  overtake  him  ? 
Let  a;=the  number  of  hours  the  2d  travels; 
thena;+5=  "         "  "         "    1st      " 

.•.  12a;=:the  number  of  miles  the  2d       " 
and7(rr+5)=  "        "  "         "    1st     " 

But  by  the  supposition  the  couriers  both  travel  the  same 
number  of  miles ; 

.-.  12a;=7  (rr+5), 
12a;=7a:+35. 
Transposing,    12a;-- 7a; =3  5, 
5a;=:35, 
•  a;  =:7,  the  number  of  hours  the  second 

coirier  is  in  overtaking  the  first. 

Prob.  3.  In  a  railway  train  15  passengers  paid  £3  12*.; 
the  fare  of  the  first  class  being  6.5.,  and  that  of  the  second  4«. 
How  many  passengers  were  there  of  each  class  ? 

Let  a;=the  number  of  passengers  of  the  1st  class, 
then  15— a;=  "         "    ^  "  "      2d       " 

.•.  6a; = sum  in  shillings  paid  by  1st  class  passengers, 
and4(15-a;)=  "  "^^  "       2d    «  '' 


so  ALGEBEA. 

But  these  two  sums  amount  to  £3  125.,  or  to  12s, 
/.  6x+4{15-x)=:72, 
6a;+60-4a;rr:72. 
By  transposition,  6a; — 4a; = 72 — 60, 
2a;r=12! 

,'.  x~6  No.  of  1st  class  passengers  ; 
/.  the  number  of  2d  class  passengers=15— a;=9. 

Prob.  4.  What  number  is  that  to  which  if  6.  be  added  twice 
the  sum  will  be  24  ?  A7is,  6. 

Prob.  5.  What  two  numbers  are  those  whose  difference  is 
6,  and  if  12  be  added  to  4  times  their  sum,  the  whole  will  be 
60  ?  Ans,  3  and  9. 

Prob.  6.  Tea  at  6s.  per  lb.  is  mixed  with  tea  at  4.5.  per  lb., 
'  and  16  lbs.  of  the  mixture  is  sold  for  £3  18^.     How  many 
lbs.  were  there  of  each  sort  ?  Ans,  1  lbs.  and  9  lbs. 

Prob.  7.  The  speed  of  a  railway, train  is  24  miles  an  hour, 
and  3  hours  after  its  departure  an  Jm)ress  train  is  started  to 
run  32  miles  an  hour.  In  how  m/nySiours  does  the  express 
overtake  the  train  first  started  1      '  Ans,  9  hours. 

Prob.  8.  A  mercer  having  cut  19  yards  from  each  of  three 
equal  pieces  of  silk,  and  17  from  another  of  the  same  length, 
found  that  the  remnants  taken  together  measured  142  yards. 
What  was  the  length  of  each  piece  1 

Let  a;=the  length  of  each  piece  in  yards  ; 
.*.  a;— 19=the  length,  of  each  of  the  3  remnants, 
and  a;— ]7=the  length  of  the  other  remnant ; 
then  3  (a;-19)+^-n=:142, 
or3a;-57+a;-17=:142, 

4a;-74=:142.  • 

Transposing,  4a; = 1 42 + 74, 

4a;=z216; 
/.  a; =54. 

Prob.  9.  Divide  the  number  68  into  two  such  parts,  that 
the  difference  between  the  greater  and  84  may  equal  3  times 
the  difference  between  the  less  and  40. 
Let  a;=the  less  part, 
then  68— a;=the  greater; 


SIMPLE   EQUATIONS.  31 

.'.  84  — (68— a;)= difference  between  84  and  the  greater, 
and  3  .  (40— a;)  =3  times  the  difference  between  the  less  and 
40. 

But  by  the  question  the  differences  are  equal  to  each  other ; 
...84-(68-ir)=J8  .  (40-^), 
or  84-684-^=120-3^;. 
By  transposition,     a;+3a;z=120-{-68— 84, 

.-.  rr=26,  the  less  part ; 
and  .'.  the  greater z=  42. 

Prob.  10.  a  man  at  a  party  at  cards  betted  three  shillings 
to  two  upon  every  deal.  After  twenty  deals  he  won  five  shil- 
lings.    How  many  deals  did  he  win  ? 

Let  x=zthe  number  of  deals  he  won ; 
.*.  20— a;=zthe  number  he  lost; 
.-.  2^  — the  money  won  ; 
and  3  .  (20— .1;)=:  the  money  lost. 

But  the  difference  between  the  money  won  and  the  money 
lost  was  55. 

.'.2x"-S.{20—x)z=  5,  • 

2a;— 60+3a;=  5, 

6x-60=  5, 

5.r— 65; 

.-.  x=lS. 

Prob.  11.  A  and  B  being  at  play  cut  packs  of  cards  so  as 
to  take  off  more  than  they  left.  Now  it  happened  that  A  cut 
off  twice  as  many  as  B  left,  and  B  cut  off  seven  times  as  many 
as  A  left.     How  were  the  cards  cut  by  each  1 

Suppose  A  cut  off  2x  cards ; 
then  52— 2a:  =:  the  number  he  left, 
and  a;=the  number  B  left; 
.*.  52— or  =: the  number  he  cutoff. 
But  the  number  B  cut  off  was  equal  to  7  times  the  number 
A  left; 

.-.  52-.t;=:7  .  (52-2a;) 
52~-x=:S64-Ux. 
Transposing,  1 4a; — a:  =:  364 — 52, 
13a;z:z312; 
.-.  a;=i24; 
.-.  A  cut  off  48,  and  B  cut  off  28  cards. 


/ 


32  *      ALGEBRA. 

Prob.  12.  Some  persons  agreed  to  give  sixpence  each  to  a 
waterman  for  carrying  them  from  London  to  Greenwich ;  but 
with  this  condition,  that  for  every  other  person  taken  in  by 
the  way,  threepence  should  be  abated  in  their  joint  fare. 
Now  the  waterman  took  in  thret  more  than  a  fourth  part  of 
the  number  of  the  first  passengers,  in  considerr.tion  cf  which 
he  took  of  them  but  fivepence  each.  How  many  persons  were 
there  at  first  1 

Let  4x  represent  the  number  of  passengers  at  first ; 
then  S  more  than  a  fourth  part  of  this  number = a; +3,  and 
they  paid  3  (x+S)  pence. 

.-.  the  original  passengers  paid  6x4rr~3  (x+S)  pence. 
But  the  original  passengers  paid  5x4a;  pence  ; 
.*.  by  equalizing  these  two  values,  we  get 
6x4:r~3(a;+3)z=:5x4(r, 
24:r  — 3:?;— 9=20;^. 
Transposing,       24x — 3^ — 20:r  =r  9  ; 

and  .'.  the  No.  of  passengers  were=:4x9=36. 

Prob.  13.  There  are  two  numbers  whose  diflTerence  is  14, 
and  if  9  times  the  less  be  subtracted  from  6  times  the  greater, 
the  remainder  will  be  33.     What  are  the  numbers  ? 

Ans.   17  and  31. 

Prob.  14.  Tw^o  persons,  A  and  B,  lay  out  equal  sums  of 
money  in  trade ;  A  (/ains  £120,  and  B  loses  £80 ;  and  now 
A's  money  is  t7'eble  of  B's.     What  sum  had  each  at  first  ? 

Ans.  £180. 

Prob.  15.  A  rectangle  is  8  feet  long,  and  if  it  were  two  feet 
broader,  its  area  would  be  48  feet.     Find  its  breadth. 

A71S,  4  feet. 
Prob.  16.  William  has  4  times  as  many  marbles  as  Thomas, 
but  if  12  be  given  to  each,  William  will  then  have  only  twice 
as  many  as  Thomas.     How  many  has  each  ? 

Ans.  24  and  6. 
Prob.  17.  Two  rectangular  slates  are  each  8  inches  wide, 
but  the  length  of  one  is  4  inches  greater  than  that  of  the  other. 
Find  their  lengths,  the  longer  slate  being  twice  the  area  of  the 
other. 

Let  a;— the  length  in  inches  of  the  less ; 
then  a; -I- 4=  "       "  "  "     greater. 


SIMPLE   EQUATIONS.  33 

Now  the  area  of  a  rectangle  is  its  length  multiplied  by  its 
breadth ; 

.'.  Sx  and  8  (rr+4)  are  the  areas  of  the  slates. 
But  the  larger  slate  is  twice  the  area  of  the  less ; 
.'.SxX2=:S{x+4:),^ 

.-.  8a:=32; 
.-.  x=z  4,  the  length  of  the  less  slate, 
and  0:4-4=  8,    "         "         "      greater  slate. 
Prob.  18.  Two  rectangular  boards  are  equal  in  area;   the 
breadth  of  the  one  is   18  inches,  and  that  of  the  other  16 
inches,  and  the  difference  of  their  lengths  4  inches.     Find  the 
length  of  each  and  the  common  area. 

Ans.  32,  36,  and  576. 

Prob.  19.  A  straight  lever  (without  weight)  supports  in 
equilibrium  on  a  fulcrum  24  lbs.  at  the  end  of  the  shorter  arm, 
and  8  lbs.  at  the  end  of  the  longer,  but  the  length  of  the  longer 
arm  is  6  inches  more  than  that  of  the  shorter.  Find  the 
lengths  of  the  arms. 

Let  a;= length  in  inches  of  the  shorter  arm  ; 
then  a; 4- 6=     "  "  "       longer       " 

Now  the  lever  will  be  in  equilibrium,  when  the  weight  at 
one  end  multiplied  by  the  length  of  the  corresponding  arm  is 
equal  to  the  weight  at  the  other  end,  multiplied  by  its  corres- 
ponding arm ; 

.\24x=S{x-\-Q), 
24:r=i8x+48, 
^16a:=48; 
.'.  x=S  inches,  the  length  of  the  shorter  arm  ; 
and  a:  4- 6 =9      "         "       "         "       longer     « 
Prob.  20.  A  weight  of  6  Jbs.  balances  a  weight  of  24  lbs.  on 
a  lever  (supposed  to  be  without  weight),  whose  length  is  20 
inches ;  if  3  lbs.  be  added  to  each  weight,  what  addition  must 
be  made  to  each  arm  of  the  lever,  so  that  the  fulcrum  may 
preserve  its  original  position,  and  equilibrium  still  be  re- 
tained 1 

This  problem  resolves  itself  into  two  other  problems  : — 

(1.)  To  find  the  lengths  of  the  arms  in  the  original  posi- 
tion : 

Let  a:=the  length  in  inches  of  the  shorter  arm ; 
then  20— a:  =  "        "  "  "      longer     " 


34  ALGEBRA. 

Now,  in  order  that  there  may  be  equilibrium,  24a;  and  6  (20 
— x)  must  be  eq^ual  to  each  other ; 

30a:  ==120; 

,\  xz=     4,  the  ^ngth  of  the  shorter  arm; 
and20-a;=  16,    "   ^    "         "      longer     " 

(2.)  To  find  the  addition  to  be  made  to  each  arm,  so  that 
there  may  again  be  equilibrium  on  the  fulcrum  in  its  original 
position,  after  3  lbs.  have  been  added  to  each  weight : 

Let  X  =  .number  of  inches  to  be  added  to  each  arm  ; 
then  the  lengths  of  the  arms  become  4+a?  and  16+a;  inches 
respectively :  and  the  weights  at  the  arms  have  been  respect- 
ively increased  to  27  lbs.  and  9  lbs. 

But  by  the  principle  of  the  equilibrium  of  the  lever,  27  (4 
+ar)  and  9  (16+^)  niust  be  equal  to  each  other; 
/.  27(4+0;)  =9(16+0:). 
Divide  each  side  of  the  equation  by  9,  and 
3(4+0:)  =  16+0:, 
12+3o:  =  16+o;,       . 
3o:-o:  =  16-12 
2o;=i  4; 
,\x=:  2. 

Prob.  21.  The  condition  being  the  same  as  in  the  last 
problem,  how  many  inches  must  be  added  to  the  shorter  arm 
in  order  that'  the  lever  may  in  its  original  position  retain  its 
equilibrium  1  Ans,  1^  inch. 

Prob.  22.  A  garrison  of  1000  men  were  victualled  for  30 
days  ;  after  10  days  it  was  reinforced,  and  then  the  provisions 
were  exhausted  in  5  days ;  find  the  number  of  men  in  the  re- 
inforcement. Alls,  3000. 

Prob.  23.  Two  triangles  have  each  a  base  of  20  feet,  but 
the  altitude  of  one  of  them  is  6  feet  less  than  that  of  the  other, 
and  the  area  of  the  greater  triangle  is  twice  that  of  the  less. 
Find  their  altitudes.  Ans.  6  and  12. 

N.  B.  The  area  of  a  triangle  =  ^  base  X  altitude. 

Prob.  24.  A  and  B  began  to  play  with  equal  sums;  A 
won  125. ;  then  6  times  A's  money  was  equal  to  9  times  B's. 
What  had  each  at  first  1  Ans,  £S. 


^  SnVIPLE   EQUATIONS.  85 

Prob.  25.  A  company  settling  their  reckoning  at  a  tavern, 
pay  4  shillings  ea^h,  but  observe  that  if  thpre  had  been  5 
more  they  would  only  have  paid  3  shillings  each.  How 
many  were  there  ?  Ans.  1^, 

Prob.  26.  Two  persons,  A  and  B,  at  the  same  time  set 
out  from  two  to wn^^a^ ja ilea  apart,  and  meet  each  other  in 
5  hours,  but  B  walks  ^N|^te|^U||^fl|||ore  than  A.  How 
many  miles  does  A  walk  inaSHKH^      Ans,  3  miles. 


DIVISION. 

36.  The  Division  of  algebraic  quantities  is  the  finding  their 
quotient,  and  in  performing  the  operation  the  same  circum- 
stances are  to  be  taken  into  consideration  as  in  their  multipli- 
cation^ and  consequently  the  four  following  Rules  must  be 
observed. 

(1.)  That  if  the  signs  of  the  dividend  and  divisor  be  liJce^ 
then  the  sign  of  the  quotient  will  be  +  ;  if  unlike^  then  the 
sign  of  the  quotient  will  be  -—  .* 

(2.)  That  the  coefficient  of  the  dividend  is  to  be  divided  by 
the  coefficient  of  the  divisor^  to  obtain  the  coefficient  of  the 
quotient. 

(3.)  That  all  the  letters  common  to  both  the  dividend  and 
the  divisor  must  be  rejected  in  the  quotient, 

(4.)  That  if  the  same  letter  be  found  in  both  the  dividend 
and  divisor  with  different  indices,  then  the  index  of  that  letter 

*  The  Rule  for  the  signs  follow  immediately  fi-om  that  in  Multipli- 
cation; thus, 


If  Ji^aX-\'h—-\-ah,  then  ^  =-f  6,  and  4^  =+a 

4  «  X — 6= — ah,  -  -  -  - —  = — 6,  and  — -  =+a 
-}-  a  —  0 

— aX — 6=+a6,  -  -  -  — —  = — 6,  and  X^  = — a 
—  a  — b 


i.e^  like  signs 
"  produce^-,  and:    t 
unlike  signs  — i,  >  I 


86.  "What  is  meant  by  the  division  of  algebraic  quantities  ?    State  th» 
Rules  in  Division. 


36 


ALGEBKA. 


in  the  divisor  must  be  subtracted  from  its  index  in  the  dividend, 
to  obtain  its  index  in  the  quotient.     Thus, 

(1.)  +abc  divided  by  +ac  -  - 


-\-ac 
(2.)  +6a6c -2a  -  -  or     -^- 

— lOxyz 


(3.)  --lOxj/z 


'  +5y-  -  or- 


+  5y 


=+b, 
=  -35c. 

■=z—2xz. 


20a'^xV 

(4.)   '-20a^xy    -  -  -  — 4aary  or——- =+5ax]/\ 

Of  Division,  also,  there  are  three  Cases :  the  same  as  in 
Multiplication, 

Case  I. 
37.  When  dividend  and  divisor  are  both  simple  terms. 
Ex.  1. 
Divide  18ax^  by  Sax, 


=6a?. 


18ax'' 
Sax 

Ex.  3. 

Divide  —  28a;^2/^  by  —  4a:y. 

Ex.5. 

Divide  —  14a^6^{j  by  7ac 
^UaWc_ 
+7ad  " 


Ex.2. 

Divide  ISa'i^  by— 5a. 

+  15a^62 


—  oa 

Ex.  4. 

Divide  25a^c^  by  — 6a^c, 
+  25aV_ 
—  5a^c      """ 

Ex.6. 

Divide  —  20^;^^^  by  —  4y2r. 


~4y^. 


Case  II. 

88.  When  the  dividend  is  a  coinpoiind  quantity,  and  the 
-divisor  a  simple  one ;  then  each  term  of  the  dividend  must  be 
.divided  separately,  and  the  resulting  quantities"  will  ic»e  the 
-quotient  required. 


State  the  rule  for  Case  2d. 


ule 


DIVISION.  8? 

Ex.  1. 

Divide  4:2a^+Sab+l2a^  by  3a. 

—- =  lAa+b+4a, 

,3a 

Ex.2. 

Divide  90aV  —  18a:z;'+4a'^—2a^  by  2aa?. 

ma^x'-lSax^+4a'x-2ax     ^^     ,     ^    ,  o       i 

J z=z4,6ax^—9x+2a-^i. 

2ax 

Ex.  3. 

Divide  Ax^ -2x'' -\-2x  by  2a;, 
4^— 2:r^+2a; 


2x 

Ex.  4.     . 

Divide  —24aVy'-^axy+QxY  by  —  3ary. 
— 24a'a:  V — ^axy + 6a;  V^__ 

Ex.  5. 
Divide  14a6^+7a^6^-21a^5H35a^6  by  ^ah, 
14a53^7^252_21a^6^+35a^6_ 

Case  III. 

39.  When  dividend  and  divisor  are  both  compound  quan- 
tities. In  this  case,  the  Eule  is,  "  To  arrange  both  dividend 
and  divisor  according  to  the  powers  of  the  same  letter,  begin- 
ning with  the  highest ;  then  find  how  often  the  first  term  of 
the  divisor  is  contained  in  the  first  term  of  the  dividend,  and 
place  the  result  in  the  quotient ;  multiply  each  term  of  the 
divisor  by  this  quantity,  and  place  the  product  under  the  cor- 
responding (i.  e.,  like)  terms  in  the  dividend,  and  then  subtract 
it  from  them ;  to  the  remainder  bring  down  as  many  terms 
of  the  dividend  as  will  make  its  number  of  terms  equal  to 

39.  When  dividond  and  divisor  are  both  compound  quantities,  what  is 
the  Kule  ? 


33  ALGEBRA. 

that  of  the  divisor;  and  then  proceed  as  before,  till  all  the 
terms  of  the  dividend  are  brought  down,  as  in  oommon 
arithmetic." 


Ex.  II 

Divide  a^Sa'b+Sab''--b'  by  a—b. 

a^b]  a'^Sa'b+Sab'-'b'  (a^^2ab+b^ 

*  -2a^6+3a5^ 


•^2a^-\-2ab' 


ab'-P 
ab'-b^ 


In  this  Example,  the  dividend  is  arranged  according  to  the 
powers  of  a,  the  first  term  of  the  divisor.  Having  done  this, 
we  proceed  by  the  following  steps : — 

(1.)  a  is  contained  in  a\  a^  times;  put  this  in  the  quo- 
tient. 


(2.)  Multiply  a— 5  by  a\  and  it  gives  a^—a^b. 

(3.)  Subtract  a^—a'b  from  a^—Sa'b,  a 
■2a'b. 

(4.)  Bring  down  the  next  term  +3a5^ 

(5.)  a  is  contained  in  —2a^b,  —  2a5  tij 
idtient. 

(6.)  Multiply/  and  subtract  as  before,  and  the  remainder  is 


(3.)  Subtract  a^—a^b  from  a^-~3a^^,  and  the  remainder  is 

-2a^6. 


(5.)  a  is  contained  in  — 2a^i,  —  2a5  times  ;   put  this  in  the 
quotient. 


aJb\ 

(7.)  Bring  down  the  last  term  —  6^ 

(8.)  a  is  contained  in  a6^,  +S^  times  ;   put  this  in  the  quo- 
tient. 

(9.)  Multiply  and  subtract  as  before,  and  nothing  remains ; 
the  quotient  therefore  is  a^—2ab'\-b'^. 


DIVISION.  39 

Ex.  2. 

)a'-\-2a^x-{-  aV    -.^  \ 


^ 


3a'ai+9aV4-10aV 

"*        3a  V+  la'x' + 5aar^ 
3a  V+  6aV+3aa;^ 

aV+2aa:^+a:^ 


/12^^~21rz;*  \ 


Ex.3. 


*~-20a;3+39a;2 

-20a:^+35a;^ 

*     +~4? 


Ex.  4. 

3a:-6\6a;*-96    /2;r^+4a;^+8a;+16 
/6^^--22^\ 

+  12^^— 24a;-^ 

*~+24?-96 
+24a;^-48a; 

*  "+48^-96 
+48a;-96 


•  "When  there  is  a  remainder,  it  must  be  made  the  numerator  of  a 
Fraction  whose  denominator  is  the  divisor  ;  this  Fraction  must  then  be 
plac(^d  in  the  quotient  (with  its  proper  sign),  the  same  as  in  common 
arithmetic. 


40  ALGEBRA. 

Ex.  5. 


a^+x-l)x'-x'+x^---x'+2x'-l(x^-^x''+x'-^x+l 
/x'+x'-x'  V 


^—x^-\-x' 
—x^—x' 

'■\-x^ 

% 

-x^-^2x—\ 
—x^—  x^+x 

*     +  x'+x- 
+  x'+x- 

-1 
-1 

*     * 

* 

A 


Ex.  6. 


-ix]x'-^x'+'-lx'--ix(x'-lx+ 1 


+  x'-ix 


^V 


Ex.  7.  Divide  a^+4a^b+6a%^+4:ab^+b^  by  a+5. 

Ex.  8.  Divide  a'— 5a^^+10aV  — 10aV+5aa:*— a;^ 
by  a^—Sa'x+Sax^-x^ 

Ans,  a^—2ax+x\ 

Ex.  9.  Divide  25x'—x*—2x^Sx^  by  5;z;^— 4a;l 

^W5.  5x^+4x^+3x+2. 

Ex.  10.  Divide  a*+8a^ar+24aV+32a^^+ 16^*  by  a+2ar. 

u4n5.  a^-}-(ja'x-^l2ax''  +  8x\ 


DIVISION.  •  ^  1 

Ex.  11.  Divide  a^—x^  by  a— ar. 

Ex.  12.  Divide  Qx'^+9^—20x  by  Zx^'-^x, 

^Ans.  2x^+2x-^^-        ^"^ 


Zx^'-Zx. 


Ex.  13.  Divide  9.c«-46a;^+ 95:^2 +150.r  by  a;^-4^-5. 

Ans.  9;?;"— 10.r^+5a;'— 30^. 

Ex.  14.  Divide  aj^~^a;^+a;^+|a?-2  by  ^x-2. 

Ans,  f.i!^— ^rr^  +  l. 


PROBLEMS     PRODUCING     SIMPLE     EQUATIONS,    CONTAINING     ONLT 
ONE    UNKNOWN    QUANTITY. 

Prob.  1.  A  fish  was  caught,  the  tail  of  which  weighed  9 
lbs. ;  his  head  weighed  as  much  as  his  tail  and  half  his  body, 
and  his  body  weighed  as  much  as  his  head  and  tail.  What 
did  the  -fish  weigh  ? 

Let  2ir= weight  of  the  body  in  lbs. ; 
.'.  9+^= weight  of  tail+l-  body = weight  of  head. 
But  the  body  weighs  as  much  as  the  head  and  tail ; 

,\2a;=(9+a;)+9,  ^ 

2a;=rr+18; 
.•.a;=18, 
and  .*.  2a;i=36,  the  weight  of  body  in  lbs., 
9+.'i:=27,  the  weight  of  head  in  lbs., 
and  the  weight  of  fish=3G4-27+9=721b§. 

Prob.  2.  A  servant  agreed  to  serve  for  £8  a  year  and  a 

livery,  but  left  his  service  at  the  end  of  7  months,  and  received 

only  £2  135.  4i  and  his  livery  ;  what  was  its  value? 

Let  12a;=the  value  of  the  livery  in  d. 

But  £8  =  1920c?.,  and  £2  13^.  4c^.=:640d  ; 

then,  the  wages  for  12  months =12^* +1^20  ; 

^     .           1      12a;+1920 
.'.  the  wages  for  1  month = — =:a;+loO. 

and  .'.  the  wages  for  7  months = (a; +160)  7. 
4* 


42 '  ALGEBRA. 

But  the  wages  actually  received  for  7  months =12a?-f-640 ; 
/.  12:r+640=:7ar  +  1120; 

.-.  5a:=480, 

and  .-.  12x=zUb2d.=£4:  16s.,  value  of  livery. 

From  the  solutions  of  the  two  preceding  problems,  it  will 
be  seen,  that  by  assuming /or  the  unknown  quantity^  x  with  a 
'proi^r  coefficient,  an  equation  free  from  fractions  will  be 
obtained.  It  is  frequently  not  only  convenient  to  make  such 
an  assumption,  but  a  more  elegant  solution  is  generally 
thereby  obtained.  The  coefficient  of  x  must  be  a  multiple 
of  the  denominator  of  all  the  fractions  involved  in  the 
problem. 

Prob.  3.  A  cistern  is  filled  in  20  minutes\  by  3  pipes,  one 
of  which  conveys  10  gallons  more,  and  another  5  gallons  less, 
than  the  third  per  minute.  The  cistern  holds  820  gallons. 
How  much  flows  through  each  pipe  in  a  minute  ? 

Ans,  22,  7,  and  12  gallons,  respectively. 

Prob.  4.  A  and  B  have  the  same  income  :  A  lays  by  \  of 
his  ;  but  B  spending  £60  a  year  more  than  A,  at  the  end  of  4 
years  finds  himself  £160  in  debt.  What  did  each  annually 
receive  1  Ans,  £100, 

Prob.  5.  A  met  two  beggars,  B  and  C,  and  having  a  certain  ' 
sum  in  his  pocket,  gave  B  -^  of  it,  and  C  f  of  the  remainder: 
A  now  had  20d  left;  what  had  he  at  firsts  Ans,  5s. 

Prob.  6.  A  person  has  two  horses,  and  a  saddle  worth 
£60 :  if  the  saddle  be  put  on  the  first  horse,  his  value  will  be- 
come double  that  of  the  second ;  but  if  it  be  put  on  the  second, 
his  value  will  become  triple  that  of  the  first.  What  is  the 
value  of  each  horse?  A7is,  £36  and  £48. 

Prob.  7.  A  gamester  at  one  sitting  lost  I  of  his  money, 
and  then  won  ISs. ;  at  a  second  he  lost  -J  of  the  remainder, 
and  then  won  Ss.,  after  which  he  had  3  guineas  left.  How 
much  money  had  he  at  first  1 

Let  15.r=the  number  of  shillings  he  had  at  first; 
having  lost  I  of  his  money,  he  had  f  of  it,  or  12.r  remaining ; 
he  then  won  I85.,  and  therefore  had  12a; -f  18  in  hand ;   losin,^ 


ALGEBRAIC   FRACTIONS.  43 

J  of  this,  he  had  f  of  it,  or  8^'  + 12  left ;  he  then  won  Ss.,  and 
so  had  {Sx-{-12)-{-S  sliillings,  which  was  to  be  equal  to  3 
guineas,  or  63^. 

.-.  (8a;+12)+3=63, 
8a;+12=r6(f; 

x=  6', 
Hence  15x=9ps.=£4:  lOs. 

Prob.  8.  A  person  at  plaj;  lost  a  third  of  his  money,  and 
then  won  45. ;  he,  again  lost  a  fourths  of  his  money,  and  then 
won  135. ;  lastly,  he  lost  an  eighth  of  what  he  then  had,  and 
found  he  had  285.  left.     What  had  he  at  first  ? 

<^    ^^       ^    ^""^ 

-K  ^      TN    OHAPTEE    III. 

^-   \       ij*  ON    ALGEBRAIC     FRACTION. 

40.  The  Rules  for  the  management  of  Algebraic  Fractions 
are  the  same  as  those  in  common  arithmetic.  The  principles, 
on  which  the  rules  in  both  sciences  are  established,  are  the 
following : — 

(1.)  If  the  numerator  of  a  fraction  be  multiplied,  or  the  de- 
nominator divided  by  any  quantity,  the  fraction  is  rendered 
so  many  times  greater  in  value. 

(2.)  If  the  numerator  of  a  fraction  be  divided,  or  the  de- 
nominator multiplied  by  any  quantity,  the  fraction  is  rendered 
80  many  times  less  in  value. 

(3.)  If  both  the  numerator  and  denominator  of  a  fraction 
be  multiplied  or  divided  by  any  quantity,  the  fraction  re- 
mains unaltered  in  value. 


40.  Are  the  same  rules  employed  in  the  management  of  algebraic  frac- 
tions as  in  those  of  common  arithmetic?  Enuraerato  the^principles  which 
are  the  foundation  of  the  rules  in  both  sciences. 


44  ALGEBRA. 

ON    THE    REDUCTION    OF    FRACTIONS. 

41.   To  reduce  a  mixed  Quantity  to  an  improper  Fraction, 

Rfle.  "  Multiply  the  integMl  part  by  the  denominator  of 
the  fraction,  and-to  the  product  annex  the  numerator  with  its 
proper  sign ;  under  this  sum  place  the  former  denominator, 
and  Ihe  result  is  the  improper  fraction  required." 

Ex.  1. 

2x 
Eeduce  8«^+^2  to  an  improper  fraction. 

The  integral  part  X  the  denominator  of  the  fraction  +  the 
nwnerator = 3a  X  5a^ + 2a; = 1 5a^ + 2x  ; 

Hence, — ^ —  is  the  fraction  required. 

Ex.  2. 

4a? 

Reduce  ^x—  —  to  an  improper  fraction. 

Here  5a;X6a^=30a^ir ;  to  this  add  the  numerator  with  its 
proper  sign,  viz.,  —4a:;  then  -—^ —  is  the  fraction  re- 
quired. 

^     Ex.  3. 
Reduce  5a; ~ —  to  an  improper  fraction. 

Here    5a;  X  "7= 35a:.      In    adding    the    numerator   2.2:— 3 

with  its  proper  sign^  it  is  to  be  recollected,  that  the  sign  — 

2a: 3 

affixed  to  the  fx-action  — — —  means  that   the  whole  of  that 

7 

fraction  is  to  be  subtracted^  and  consequently  that  the  signs 
of  each  term  of  the  numerator  must  be  changed  when  it  is 
combined  with  35a? ;  hence  the  improper  fraction  required  is 
35a:-2a:+3     33a:+3 


41 .  How  is  a  mixed  quantity  reduced  to  an  improper  fraction  ? 


ALGEBKAIC  FRACTIONS.  45 

2c 
Ex.  4.  Reduce  4a5+— to  an  improper  fraction. 

,      A2a'b-{'2c 

4a        ^ 
Ex.  5.  Reduce  35^— —  to  an  improper  fraction. 
ox 

.        155^;r— 4a 
Ans, . 

DX 

d^ — ax 
Ex.  6.  Reduce  a—x-{ to  an  improper  fraction. 

a'-x^ 
Ans,  . 

X 

^^ 9 

Ex.  7.  Reduce  3a;^ --—  to  an  improper  fraction. 

30^'-4^+9 


Ans, 


10 


42.   To  reduce  an  improper  Fraction  to  a  mixed  Quantity. 

Rule.  "  Observe  which  terms  of  the  numerator  are  divisi- 
ble by  the  denominator  without  a  remainder,  the  quotient 
will  give  the  integral  part ;  to  this  annex  (with  their  proper 
signs)  the  remaining  terms  of  the  numerator  with  the  denom- 
inator under  them,  and  the  result  will  be  the  mixed  quantity 
required." 

Ex.  1. 

Reduce to  a  mixed  quantity. 

Here =a+6  is  the  integral  part, 

and  —  is  the  fractional  part ; 

b^  , 

/.  a+^H —  is  the  mixed  quantity  required. 

Si . 

42.  What  is  the  rule  for  reducing  an  improper  fraction  to  a  mixed  quan" 

tity?  ^ 


46  ALGEBRA. 

Ex.  2. 

Keduce to  a  mixed  quantity. 

Here  — — =Sa  is  the  i?itegrS -pa,rtf 

2^ g^ 

and  — - —  is  the  fractional  part ; 
oa 

2x—Sc  .     ,        .     , 
/.  3a  H —  IS  the  mixed  quantity  required. 

Ex.  3.  Reduce  — to  a  mixed  quantity. 

Ans.  2x—^r-, 
2x 

Ex.  4.  Eeduce ^ ^^  *  mixed  quantity. 

^a 

3c 

^n5.  3a+l— —-. 
4a 

Ex.  5.  Reduce — -^ to  a  mixed  quantity. 

X 

2b^ 
Ans,  10y4-3a; ^, 


43.   To  reduce  Fractions  to  a  common  Denominator, 

Rule.  "  Multiply  each  numerator  into  every  denominator 
hut  its  own  for  the  new  numerators,  and  all  the  denominators 
together  for  the  common  denominator." 

Ex.  1. 

2'X   ^x  4a 

Reduce  -^j  -y,  and  — ,  to  a  common  denominator. 
o      o  o 

Hence  the  frac- 
tions required  are 
lObx    75^    12a^ 
1X56"'    156'    156* 


2xXbX^  =  lObx^ 

5:c  X  3  X  5 = 15x    y  new  numerators  ; 

4aX3x6  =  12a6j 

3   X6X5=156  common  denominator ; 


43.  How  are  fractions  reduced  to  a  common  denominator  ? 


ALGEBRAIC   FRACTIONS.  47 

Ex.2. 

Keduce  — - — ,  and  -— ,  to  a  common  denomuiator. 
5  4 


Hence   the   frac- 
tions required 
are 
8:^^+4        ,   Ibx 
-20"'  ^^^    20-' 


f 

Here  (2^+1) X4=  8^+4)    new  nume- 
3a;  X  5  =  1 5^        )       rators  ; 

5X4=20  common  denomi- 
nator ; 

Ex.3. 

Eeduce  — ; — ,  -^r— ,  and  r-,  to  a  common  denominator; 
a-\-x      3  2x 

Here  5xX  Sx2x=S0x^  ]  •*•  the  new  frac-       30x^ 

{a-^x)     {a-{-x)x2x=2a''x—2x^[  tions  are     6ax+6x^' 

1      X(a+^)X3  =Sa    +3.r    1 2a'x-2x^  Sa+Sx 

{a-\-x)  X3  X2:i:        =6ax^6x''J  Qax -i-6?'  6^+0?' 

337  4:bx  5x 

Ex.  4.  Reduce  -— .  -^r-,  and  — ,  to  a  common  denominator. 
o     oa  a 

9a'x  20abx       ,  75ax^ 

■n^     n    -n  ;,        2a;  +  3       ^  5a:+l 

JbiX.  5.  Keduce ,  and  — — — ,  to  a  common  denommator. 

X  o 

.        6a;+9       ,  5a;2+a; 

Ans,  —- — ,  and  -— , 

Sx  Sx 

Ex.  6. 

4:X^4"2x    Sx^  2x 

Reduce — ,  -— ,  and  — ,  to  a  common  denominator. 

o  4a  oo 

.        4:Sabx''+24ahx  45bx''       ,  4.0ax 

Ex.  7. 

7x'^—l  4x^—x-{-2 

Reduce  — - — ,  and — ^ — ,  to  a  common  denominator. 

2x  2a^ 

14aV~2a2  8^^-2a:^+4^ 


i8  ALGEBRA. 

44.   To  reduce  a  fraction  to  its  lowest  terms. 

Rule.  "  Observe  what  quantity  will  divide  all  the  term3 

both  of  the  numerator  and  denominator  without  a  remainder ; 

Divide  them  by  this  quantity,  |nd  the  fraction  is  reduced  to 

its  lowest  terms." 

Ex.  1. 

^   ,       \4:r^-\-lax+2lx'      .     , 

Keduce --—- to  its  lowest  terms. 

The  coefficient  of  every  term  of  the  numerator  and  denomi- 
nator of  this  fraction  is  divisible  by  7,  and  the  letter  x  also 
enters  into  every  term ;  therefore  Ix  will  divide  both  nume- 
rator and  denominator  without  a  remainder. 

Now '—2x^-{-a+^x, 

Ix 

^  Zbx^     , 
and-— — =5^; 

Hence,  the  fraction  in  its  lowest  terms  is . 

5a; 

Ex.  2. 

^   ,        20a5c— 5a^+10ac       .     , 

Keduce — -r to  its  lowest  term. 

ha^c 

Here  the  quantity  which  divides  both  numerator  and  de- 
nominator without  a  remainder  is  5a;  the  fraction  therefore 

.    .     ,  .    46c— a+2c 

m  its  lowest  terms  is . 

ac 

Ex.  3. 

Reduce  — — r-„  to  its  lowest  terms. 
a^—¥ 

Here  a—h  will  divide  both  numerator  and  denominator, 

for  by  Ex.  2.    Case  III.   page  27.   a^^y'^{a+b)  (a-5); 

hence  — — r-  is  the  fraction  in  its  lowest  terms. 
a-\-b 

\()x^ 

Ex.  4.   Reduce  ——-^  to  its  lowest  terms. 
Ibx^ 

2x 

^^^-    3 ' 

4A.  Show  how  fractions  are  reduced  to  their  lowest  terms. 


ALGEBRAIC  FRACTIONS.  49 

Ex.  5.   Eeduce  -r: —  to  its  lowest  terms. 
Qax 

.        hx 
Ans.  -. 

UxV 21:c^y^ 

Ex.  6.  Eeduce r— ^ to  its  lowest  terms. 

"Ix^y 

Ans.  ^I^:^. 

X 

Ex.  7.  Reduce — — to  its  lowest  terms. 

Yix^ 

Sx''-x^\-2 
Ans,  -, . 


ON     THE    ADDITION,     SUBTRACTION,     MULTIPLICATION,     AND 
DIVISION,    OF    FRACTIONS. 

45.   To  add  Fractions  tog-ether. 

Rule.  "  Reduce  the  fractions  to  a  common  denominator 
and  then  add  their  numerators  together;  bring  the  result- 
ing fraction  to  its  lowest  terms,  and  it  will  be  the  sum  re- 
quired." 

Ex.  1. 

Add  —,-=:,  and  -,  together. 

O      7  o 

3a:X7x3=63a;] 

2a:X5x3=30a;  (      6SxA-S0x+S5x      128.i;   .     ,      ^       .      ^ 
.X5X7:=35.  \.-- ro-5—  =  -105-  ^^  ^^T^ 


5x7x3=105. 

^     ^     .  ^  ^   a  2a         .6b             . 

Ex.  2.  Add  -,  — -,  and  -— ,  together, 

6  36            4a        ° 


aX35x4a=12a^6'' 
2aX6X4a=   Sa^b 
56X^X36  =  156^ 
6x36x4a  =  12a62j 


12a'^6  +  8a^6  +  156^  _  20a^6  +  156» 

•*•  12^6^  ~        12^2 

//I-  •;,•       T.      7^   20a^+156^   .      . 
=  (dividmg  by   6)   — 12^—   is   the 

sum  required. 


45.  Stftte  the  rule  for  adding  fractions. 


50  ALGEBRA. 

^     «     A  -,-.  2^+3  3^—1        ^4x  ^        . 
Ex.  3,  Add  — - — ,  —^ — ,  and  — ,  together. 

(2.t'  +  3)x2.rX 7=28^2  +42:c^     .   28x^+42a:+105a:— 354-40a:« 
(3^-l)x5    x7=:105:r-35    U*  70ar 

5X2^7^70^  J    """"        70^  ^^    ^  ^ 

»  .  sum  required. 

_      ^       .  ^  ^  3:c   5a;       _  4a; 

Ex.  4.    Add  — ,  — ,  and  — ,  together. 

934a; 

Arts, 


603' 


^  ■  .  , ,  3a2  2a        ,35 

Ex.  5.    Add  ^j  "H"'  ^^^^  iT"'  together. 


105a^+28a^6+306« 


70a6 


-n     ^      Ajj^^+1    4a;4-2       _  a;  ^        . 
Ex.  6.    Add  — ——^  — -— ,  and  -,  together. 

16ar+77 

^^^^-  -T05- 

Ex.  7.    Add  — ■— — ,  and  — — — ,  together. 

37a.'+115 

Ex.  8    Add  — - — ,  and  -^r-^  together. 

^''"-  —- 6^' 
Ex.  9,    Add -,  and  — -^,  together. 

X  —  *i  X-\-0 

2a;2 
Ans. 


x'-9' 


Ex.  10.  Add L,  and  — — ,,  together. 

a — 0  a-f-o 

2a^+26* 


ALGEBRAIC  FRACTIONS.  5i 

46.   To  Subtract  Fractional  Quantities, 

BuLE.  "  Reduce  the  fractions  to  a  common  denominator  ; 
and  then  subtract  the  numerators  from  each  other,  and  under 
the  difference  write  the  common  denominator." 

Ex.  1. 

Subtract  —  from  -~r-. 
5  15 

3;rXl5=45:r)       70a;— 450-      25a:       x,      ,      ..^ 

UxX  5=70a;  V  •'• 75 —  =  "75"  =  3  '^  ^^^  difference 

5  Xi5=75    )  required. 

Ex.2. 

^_  2x+l^        5x+2 

Subtract  — -—  from  — - — . 

(2a:+l)x7=14a;+7)         l5x+6-Ux-1f      x-1  ,     , 

(5a;+2)x3z:rl5a:+6  V 21 "^  l^T  '^  ^^ 

3x7=21  )    fraction  required. 

Ex.  3. 

lOo;— 9     ,^        3a;— 5 

From  — - —  subtract  — — — . 

(10a:-9)  x7=70a;-63  )        70a; -63 -24a; +40  _  46a;-23 
(  3a;-5)x8=24a;-40V   •*•  56  -       56 

8   X  7=56  )    is  the  fraction  required. 

Ex.  4. 

_         a+b      .        ^a—b 

Erom subtract  — tt. 

a— 6*  a-{-b 

^     ,  A^r     .  n       2.0  a.aO    .  a'-^'^ab-\-b'-a'+2ab--b' 
{a-{-b)  {a-]-b)=a^+2ab-}-b' 

{a-b)  {a-b)=a'^2ab+b 


(^ct-b){ai-b)=:a'-b- 


a^-b'' 
-5 — 7j  IS  the  fraction  required. 


Ex.  5.   Subtract  —  from  — .  Am.  -^^ 


46.  (5ive  the  rule  for  subtracting  fractions. 


52 


ALGEBRA. 


Ex,   6.   Subtract  — ——  from : — . 


Ex.    7.    Subtract  — -— -  from  ---. 

Ex.   8.   Subtract?:?;^  from  ^~. 
Sx  3 

Ex.   9.   Subtract  — --7  Irom  — -. 
a-f-o  a—o 

Ex.  10.   Subtract  — -^ —  from  — . 


\21x+ 17 


U^ltO, 

28       • 

4a:2_ll^_ 

-5 

50^+5 

Ans, 

4x'  +  S 

Ans. 
Arts. 

2b 
a'-b'' 

lla;+49 

56 


8 


47.   To  Multiply  Fractional  Quantities* 

KuLE.  "  Multiply  their  numerators  together  for  a  new 
numerator,  and  their  denominators  together  for  a  new  de- 
nominator, and  reduce  the  resulting  fraction  to  its  lowest 
terms." 

Ex.  1. 


2xx4:X=zSx'^ 
7   x9 


Multiply  -^  by  -|. 
=63  i   •*•  ^^^  fraction  required  is 


Sx^ 
63* 


Ex.  2. 

,^  ,  .I    4x+l  ,     6^ 
Multiply — -—  by  y. 


Here* 

{4.x+i)x6xz=z24:x^+6x  ^ 
and 

3x7  =21 


24:x^+6x 
21 


=  (dividing   the   nu- 


merator and  denominator  by  3) 
— - —  is  the  fraction  required. 


47.  State  the  rule  for  the  multiplication  of  fractio:is. 


ALGEBRAIC   FRACTIONS.  53 

Ex.  3. 

Multiply  — — ,—  Dv  — — ;. 

By  Ex.  2.    Case   III.    p^e  27,   {a'—b')xSa'={a  +  b) 

i       7x     o  9      1              1             1         .     3a'X  (a+^>)(«-^) 
(a— 6)Xoa'':    hence   the  product   is   -rr — .     \-t — ^  = 

(dividing  the  numerator  and  denominator  by  a +5) —^ - 

Ex.  4. 

Multiply  — — —   by 


14  '  '^x^'—Zx 

21a.r^— 35aa; 


Here 

(3;i;'— 5:r)  X7a=21aa;^— 35aa; 

and 
(2.r^-3aj)  X  14=28^-^-42a; 


••    28.3-42.    -^(^^-^^-g 

the  numerator  and  denonii- 

^      -      ^  .  3a.— 5a  .     , 
nator  by  7.)  — — ; — -   is  the 

fraction  required. 


Ex.  5.    Multiply by  — .  ^4/15. 


.T-i  -^  7*  •  7.-r 

T.        ^       n^r    1   .    ,       3.2  —  .    ,10  ,   ,  8.— 1 

Ex.  6.   Multiply  -^  by  ^-^,-^.  Ans.  — -. 

Ex.  7.    Multiply by  — -— .  Arts.  — - — 

Ex.  8.   Multiply  ^— ^  by  -^_ .  Ans.  -. 


48.   0/i  //i^  Division  of  Fractions, 

Rule.  "  Invert  the  divisor,  and  proceed  as  in  Multiplioa 
tion." 


48.  Enimciate  the  rule  for  division  of  fractions. 


54 


ALGEBRA. 


Ex.  1. 
Divide  -^  bj  — . 

Inter t  the  divisor,  and  it  becomes  --  :   hence  -— -  x  ;r- 

2x'  9  2a? 

42ic'^      7x 
=  r-^—  z=z  —  (dividing  the  numerator  and  denominator  by  6x) 

is  the  fraction  required. 

Ex.  2. 

^.  .,    Ux—S  ,     10x^-4: 

Divide  — z —  by 


Ux-^S  25 

— ^ —  X 


5        ^'       25     ' 
(14^_3)X5      70a;--15 


10a;-4  " 


lOx-4: 


lOx-^4. 


Ex.  s; 

Divide  — by  — -- — . 

2a  -^       6b 


ba'-6b^     5x{a+b){a-b) 

2a      ""  2a 

4a  4-46     4x(a  +  ^') 


5x{a+b){a-b)^ 


m 


6b 


66 


2a  ''4x(a+6) 

i       SObx{a-b)     15a6-156^  . 
8a  4a 

the  fraction  required. 


Ex.  4.  Divide  -x-  by  -^. 

7     -^    5 


Ex.  5.  Dunde  —7; —  by  — - — . 
3         '^      5x 


Ex.  6.  Divide  ^-  by  ^. 


Ex.  7.  Divide  ?^^  by  f-. 
3      /       5 


^/15. 


^725. 


Ans. 


Ans, 


20 


63* 

lOo; 

3  * 

4.^- 

12 

5 

9.r- 

3 

SIMPLE   EQUATIONS.  55 

ON     THE     SOLUTION     OF     SIMPLE     EQUATIONS,    CONTAINING    ONLY 
ONE    UNKNOWN    QUANTITY. 

Rule  III. 

49.  An  equation  may  be  ckared  of  fractions  by  multiply- 
ing each  side  of  the  equation  Dy  the  denominators  of  the  frac- 
tions in  succession. 

Or,  an  equation  may  be  cleared  of  fractions  by  multiplying 
each  side  of  the  equation  by  the  least  common  multiple  of  the 
denominators  of  the  fractions. 

This  Rule  is  derived  from  the  axiom  (4),  that,  if  equal 
quantities'  be  multiplied  by  the  same  quantity  (or  by  equal 
quantities),  the  products  arising  will  be  equal. 

Ex.  1.  Let  ^=6. 
Multiply  each  side  of  the  equation  by  3 ;  then  (since  the 

multiplication  of  the  fraction  ^  by  3  just  takes  away  the  de- 

o 

nominator  and  leaves  x  for  the  product)  we  have 

ir=6x3  =  18. 

Ex.  2.  Let  1+1=1. 

Multiply  each  side  of  the  equation  hy  2,  and  we  have 

a.+-=14. 

Again,  multiply  each  side  of  this  equation  by  5,  an<J        '^r 
comes  5a;+2ar=70, 

7a;=70 ; 
v.'P=10. 

Ex.3.  Let  f +  1=13- 1. 

Multiply  each  side  by  2,  then    a;+— •  =26— — . 

Multiply  each  side  by  3,  and   3a:+2a;=78  — —  . 
Multiply  each  side  by  4,  and  12a;+8ir=312— Oa:. 


56  ALGEBEA. 

By  transposition,  1 2a; + 8a:  -f  6a; = 3 1 2, 

26a:r=312; 

.-.     x=   12. 

-  This  example  might  have  been  solved  more  simply,  by  mul- 
tiplying each  side  of  the  equatmn  by  the  least  common  multi- 
ple of  the  numbers  2,  3,  4,  which  is*  12. 

Multiply  each  side  by  12,  —^ — f--^  =  156 -r^, 

or,  6a;+4a;=156— 3a;. 
By  transposition,  6a; + 4a; + 3a;— 156, 
13a;=156; 
.\x=z  12. 

Ex.  4.  Let  *^+|=22.  Ans,  x=24. 

~    Ex.5.  Let  T—-::^-?:'  ^^s.  a;=10. 

4  6      6 

Ex.  6.  Let  |+^=31-|.  Ans.  a;=30. 

Z     o  5 

Ex.  7.  Let  ~ -^+^=44.  Ans.  a;=60. 

5  o     2 

60.  In  the  application  of  the  Rules  to  the  solution  of  simple 
t/t|uations  in  general  containing  only  one  unknown  quantity,  it 
will  be  proper  to  observe  the  following  method. 

(1.)  To  clear  the  equation  of  fractions  by  Rule  III. 

(2.)  -To  collect  the  unknown  quantities  on  one  side  of  the 
equation,  txid  iliie  known  on  the  other,  by  Rule  II. 

(3.)  To  liiid  the  value  of  the  unknown  quantity  by  di- 
viding each  R^le  of  the  equation  by  its  coefficient,  as  in 
Rule  L 

_ 

50  Y.)j  r  ,-•-•;  the  three  steps  by  which  a  simple  equation  containing 
only  onw  urf  y>  .wn  quantity  may  be  solved. 


SIMPLE  EQUATIONS.  57 

Ex.  1. 

^.    -.    ,         ,        ^     .     .  .       So;       ,     a;     13 

Find  the  value  of  a;  m  the  equation  -— —  1=-+-—. 

t  o      o 

7x    91 
Multiply  b^  7,  then   Sx-\-  7=—+--. 

Multiply  by  5,  then  Ibx-^Sb-lfx+dl. 

Collect  the  unknown  quantities  on ) 

one  side,  and  the  known  on  the  >•  15a: --7a: =9 1—35, 

other ;  ) 

or  8a: =56. 

56 
Divide  by  the  coefficient  of  a:,  a:=— •  =-7. 

o 

Ex.2. 

0:4-3  X 

Find  the  value  of  a:  in  the  equation  — 1=2—-. 

5a: 
Multiply  by  5,  then   x+  3—  5  =  10——; 

Multiply  by  7,  then  7a: +21 -35=70-50:. 
Collect  the  unknown  quantities  ) 

on  one  side,  and  the  known  >•  7o:+ 5o:= 70— 21+35. 
on  the  other  ;  ) 

or  12a:=84; 
__84 

•*'^""12 
Ex.  3. 

Find  the  value  of  x  in  the  equation 

■       x-l  2a:-2  ,  _  . 

4a: ^— =^H ^ — [-24. 

2  5 

Multiply  by  the  to<  I  40^-5^+5=10.r+4^-4+240. 
common  multiple  (10),      j 

By  transposition,  40a;— 5a;— lOo:—  4a;=240— 4— 5. 
or40a:— 19.1— 231, 
i.  e.  21a:=231 ; 

.•.o:=_  =  ll. 

As  the  first  step  in  this  Example  involves  the  case  "  where 
the  sign  —  stands  before  a  fraction,"  when  the  numerator  of 


58  ALGEBRA. 

that  fraction  is  brought  down  into  the  same  line  with  40^*,  the 
signs  of  both  its  terms  must  ,be  changed^  for  the  reasons  as- 
signed in  Ex.  3,  page  44;  and  we  therefore  make  it  —  5:r-|-5, 
and  not  5a;— 5. 

Ex*4. 

X 

Find  the  value  of  a;  in  the  equation  2^—-  +1=5:^7— -2. 

Multiply  by  2,  then  Ax—X'\-2—\()x—4:. 

By  transposition,  4+2=rl0a;--4a;+^j 
or  6=  7;r; 
6  t 

6 
.  or^=-. 

Ex.  5. 

What  is  the  value  of  a;  in  the  equation  3aa;+25:r=3c+a? 

Here  3a^+26^=:(3a+26)  X:r ; 

.-.  (3a+25)x^=3c+a. 
Divide  each  side  of  the  equation  by  3a +26,  which  is  the 

coefficient  of  a; ;  then  07=- — — -,. 

3a -{-26 

Ex.  6. 
Find  the  value  of  a;  in  the  equation  Sbx+az=2ax-\-4c, 
Bring  the  miTcnown  quantities  to  one  side  of  the  equation,  and 
the  known  to  the  other  ;  then, 

Sbx—2axz=z4:C—a ; 
but  Sbx'-2ax={Sb—2a)xx; 
,\  {Sb—2a)x=4c—a. 

^Q ^ 

Divide  by  36— 2a,  and  x=— — --. 
-^  '  36— 2a 

Ex.7. 

Find  the  value  of  a;  in  the  equation  bx+x=z2x+Sa, 

Transpose  2x,  then  bx-{-x—2x=Sa,  ^ 

or  bx —   x=3a  ; 

but  6a:—  x=z{b  —  l)xi 

.-.  (6--1)  x=Sa, 

3a 
and  x=- — - . 
0 — 1 


Si^MPLE   EQUATIONS.  69 

Ex.   8.  x+l+lz=U.  Ans.  x=6. 

Ex.   9.  f+|+f=|+17^  Am.  x=QO. 

Ex.  10.   4a;-20=y+^.  Ans.  a;=10. 

-p    ,,     a;  ,  a;    a;    1  .  6 

^'^•"•2+3-4=2-  Ans..=-. 

Ex.  12.   Sx+l^"^.  Ans.  x=l. 

Sx 

'       Ex.  13.   Y-5=29--2:r.  Ans,  x=U. 

Sx 

Ex.14.   6x—--—9z=5x.  Ans.  x=B(j. 
4 

Ex.15.   2x ^+15= J- —  Ans.  a;=12. 

o  5 

^    Ex.  16.   ^+^=20-^^.  A,is.  a;=18. 
Ex.17.   5a!-?^^  +  l=3;r:+^+7.   .4»s.  x=8. 

Ex.  18.   2ax+b=2cx+4a.  Ans.  x=f^''^ 


"2a— 3c 


Ex.  19. 

,    .        .     4  7a;-9     4/  ^a;-l\     .    , 
30.-4— .^-=-(6+— j;  find  ^. 

Multiply  by  15, 45^-60-28^4-36=: 72+4^;— 4. 

45:i;-28a?-4.r— 72-4+60-36, 
13:r=:92; 


;  find  a?. 


60  ALGEBRA. 

Ex.  20. 
4a;+3     7a;— 29_8a:+19 
9     '^5a;-l^:~~18^ 

Mult,  by  18,  8a;+6+i^^^=8:r+19. 

126^522  _ 
~5^:=32"-"-^^- 
Multiply  by  5(c-12,  126a; -522==  65a: -156, 
126a:-65a;r=522-156, 
61a;=366; 
.-.  x=6, 

Ex.21. 

Mult.by7(a:-1),  7-l^ij=i^=l.    ^ 

14(a;-l) 
^"— ^7~- 
7(a:-l) 


Divide  by  2,        3=    ^^^    .^ 

3a;+21=7a;-7, 
7a;-3a;=21+7, 
4a; =28; 
•    -  .•.a;=7. 

Ex.  22. 
^      8a;+5  ,  7a;-3     16a;+15  ,  2i     .   , 

^''  -r4-+6-.+2=— 28-  +y '  ^^^  ' 

Mult.by28,16a;+10+^-— =  16a;+15+9. 
•'  ba;+2 

196^-84_ 

6a;+2    ~     ' 
196a;-84=84a;+28, 
112a;=:112; 
.•.  a;  =  l. 


SIMPLE   EQUATIONS.  *         61 

Ex.23.   — 3- =-54 J—-  ^^•^• 

•r.     c.^    9^+20     4:r-12     x  ^        a       ^ 

Ex.  24.  -^^^^-^-t^.  ^n..  8. 

Ex.  25.  -^^  +^_^^=^+^^.  ^n,.  4. 

Ex.26.  _^^-__+-^-^-_-.       ^..4. 

Ex.27.  4(5a;-3)-64(3-a:)-3(12a;-4)=96.  Ans.Q. 

Ex.28.  10(a:+|)-6^^~i^=23.      '  ^W5.  2. 

T.     or.    30+6a;  ,  60+8a;     ,,,48  .       ^ 

Ex.  29.  -^;^+-^:p3-=14+— .  An.Z. 

PROBLEMS. 

pROB.  1.  What  number  is  that  to  which  10  being  added, 
|ths  of  the  sum  shall  be  66  ? 

Let  a;=:the  number  required; 
then  a;+10=the  number,  with  10  added  to  it. 

Now  Iths  of  (.+10)=|(.+  10)=i(^tH)=?^. 

But,  by  the  question,  |ths  of  (a; +10)= 66  ; 

Hence,  — - — =66. 
5 

Multiply  by  5,  then  3iir+30=330; 

.•.3a:=330-30=300;  or  a; =^=100. 

Prob.  2.  What  number  is  that  which  being  multiplied  by 

6,  the  product  increased  by  18,  and  that  sum  divided  by  9, 

the  quotient  shall  be  20  ? 

Let  a:=the  number  required  ; 

then  6a; = the  number  multiplied  by  6; 

6a; +18= the  product  increased  by  18, 

,  6a;+18       ,  -...-,    1  n      r. 

and  * — - — =that  sum  divided  by  9. 

6 


62  *  ALGEBRA. 

Hence,  by  the  question,  — - — =20. 

Multiply  by  9,  then  6^ + 1 8  =  1 80, 

or6a;=180-fpl8  =  162;  ov  x=~=:21. 

6 

pROB.  8.  A  post  is  Jth  in  the  earth,  fths  in  water,  and  13 

feet  out  of  the  water.     What  is  the  length  of  the  post? 

Let  a;=length  of  the  post  in  ft. ; 

then  ^=the  part  of  it  in  the  earth, 

—-= the  part  of  it  in  the  water, 

13= the  part  of  it  out  of  the  water. 
But  part  in  earth  +  part  in  water  +  part  out  of  water  =: 
whole  post ; 

...     (I)     +         f^)     +  18         =.. 

15a: 
Multiply  by  5,  then  x-\ — —  +  65= 5a;; 

Multiply  by  7,  then  7x+l5x+455=:S5x, 

or  455=350?— 7rr—15a;=13ar. 

455 
Hence  a; =-—-=35  length  of  post  in  ft. 

Prob.  4.  After  paying  away  J-th  and  -^-th  of  my  money,  I 
had  £85  left  in  my  purse.     "What  money  had  I  at  first  ? 
Let  ir= money  in  purse  ^t  first ; 

then  -+-= money  paid  away. 

But  money  at  first— money  paid  away = money  remaining. 

Hence  x     —         (1+7I       =     ^^j 

1.  e.,    x—-—~:         =     85. 
>  4    7 

Multiply  by  4,  then  4:X—x — --=340; 

Multiply  by  7,  then  28a;— 7ir—4.r= 2880. 
.-.  17a;=2380; 

or  0:=-— -=£140. 

\        % 


SIMPLE  EQUATIONS.  63 

Prob.  5.  What  number  is  that,  to  which  if  I  add  20,  and 
from  |ds  of  this  sum  I  subtract  12,  the  remainder  shall  be  10  ? 

Ans,  13. 

Prob.  6.  What  number  is  that,  of  which  if  I  add  -Jd,  ^Jth, 
and  f  ths  together,  the  sum  sl^ll  be  73 '?  Ans.  84. 

Prob.  7.  What  number  is  that  whose  Jd  part  exceeds  its 
ithby72?  Ans,  540. 

Prob.  8.  There  are  two  numbers  whose  sum  is  37,  and  if 
S  times  the  lesser  be  subtracted  from  4  times  the  greater,  and 
this  difference  divided  by  6,  the  quotient  will  be  6.  What 
are  the  numbers  ?  Aris,  21  and  16. 

Prob.  9.  There  are  two  numbers  whose  sum  is  49 ;  and  if 
^th  of  the  lesser  be  subtracted  from  ^th  of  the  greater,  the  re- 
mainder will  be  5.     What  are  the  numbers  1  ^ 

Ans,  35  and#iy 

Prob.  10.  To  divide  the  number  72  into  three  parts,  so 
that  -J-  the  Jlrst  part  shall  be  equal  to  the  second,  and  |ths  of 
the  second  part  equal  to  the  third. 

Ans,  40,  20,  and  12. 

Prob.  11.  A  person  after  spending  Jth  of  his  income  plus 
£10,  had  then  remaining  ^^  of  it  plus  £35.  Required  his 
income.  A7is.  £150. 

Prob.  12.  A  gamester  at  one  sitting  lost  J-th  of  his  money, 
and  then  won  10  shillings;  at  a  second  he  lost  -J^  of  the  re- 
mainder, and  then  won  3  shillings ;  after  which  he  had  3 
guineas  left.     What  money  had  he  at  first  ?  Ans,  £5. 

Prob.  13.  Divide  the  number  90  into  four  such  parts,  that 
the  first  increased  by  2,  the  second  diminished  by  2,  the  third 
multiplied  by  2,  and  the  fourth  divided  by  2,  may  all  be  equal 
to  the  same  quantity.  Ans.  18,  22,  10,  40. 

Prob.  14.  A  merchant  has  two  kinds  of  tea,  one  worth 
9s.  6(f.  per  lb.,  the  other  13^.  6d.  How  many  lbs.  of  each 
must  he  take  to  form  a  chest  of  104  lbs.,  which  shall  be  worth 
£56?  Ans.  33  at  13^.  6d. 

71  at    95.  6d. 


64  ALGEBRA. 

Prob.  15.  Three  persons,  A,  B,  and  C,  can  separately  reap 
a  field  of  corn  in  4,  8,  and  12  days  respectively.  In  how 
many  days  can  they  conjointly  reap  the  field  ? 

Let  x  =  No.  of  days  required^ by  them  to  reap  the  field ; 
then  if  1  represent  the  work,  or  me  reaping  of  the  field, 
-J-=the  part  reaped  by  A  in  1  day, 

1__    u        a  u  B  " 

1  __    a        a  u  Q  U 

...  |+-i.4-?=:  "      "         «     all  three     " 

But  the  part  reaped  by  all  three  in  1  day  multiplied  by  the 
number  of  days  they  took  to  reap  the  field,  is  equal  to  the 
whole  work,  or  1 ; 

•••(i+KTV)«'  =  l: 
Clearing  of  fractions  by  multiplying  by  24, 
(6+3+2)  a;=24, 
lla;=24; 
.*.  x=2^^  days. 

Prob.  16.  A  man  and  his  wife  usually  drank  a  cask  of 
beer  in  10  days,  but  when  the  man  was  absent  it  lasted  the 
wife  80  days  ;  how  long  would  the  man  alone  take  to  drink 
it  1  Ans.  15  days. 

Prob.  17.  A  cistern  has  3  pipes,  two  of  which  will  fill  it  in 
8  and  4  hours  respectively,  and  the  third  will  empty  it  in  6 
hours  ;  in  what  time  will  the  cistern  be  full,  if  they  be  all  set 
a-running  at  once  ?  Ans,  2h.  24m. 

Prob.  .18.  A  person  bought  oranges  at  20d.  per  dozen ;   if 
he  had  bought  6  more  for  the  same  money,  they  would  have 
cost  4d.  a  dozen  less.     How  many  did  he  buy  ? 
Let  ojnrthe  number  of  oranges; 
then.T+6=  "         "       "         "       at  4c?.  less  per  dozen. 

Price  of  each  orange  in  1st  case=f  f  =:fd 
and  "     "      "        "       "  2d    "    =i|=|cZ. 

.*.  the  cost  of  the  oranges =—. 
o 

But  we  have  also 

the  cost  of  the  oranges =|  (a; +6). 

Two  independent  values  have  therefore  been  obtained  for 


SIMPLE   EQUATIONS.  65 

the  cost  of  the  oranges ;  these  values  must  necessarily  be  equal 
to  each  other ; 

Multiplying  each  side  of  thft  equation  by  3, 
'^5a;=4  (:rH-6), 
5a;=4a;+24; 
.-.  a; =24,  the  No.  of  oranges. 
Prob.  19.  A  market-woman  bought  a  certain  number  of 
apples  at  two  a  penny,  and  as  many  at  three  a  penny,  and 
sold  them  at  the  rate  of  five  for  twopence ;  after  which  she 
found  that  instead  of  making  her  money  again  as  she  expected, 
she  lost  fourpence  by  the  whole  business.     How  much  money 
had  she  laid  out  ?  Ans.  86\  Ad. 

Prob.  20.  A  person  rows  from  Cambridge  to  Ely,  a  dis- 
tance of  20  miles,  and  back  again,  in  10  hours,  the  stream 
flowing  uniformly  in  the  same  direction  all  the  time  ;  and  he 
finds  that  he  can  row  2  miles  against  the  stream  in  the  same 
time  that  he  rows  3  with  it.  Find  the  time  of  his  going  and 
returning. 

Let  3iP=zNo.  of  miles  rowed  per  hour  with  the  stream  ; 
.-.  2x—  "     "      "         "         "       "     against         " 
Now  the  distance  divided  by  the  rate  per  hour  gives  the  time ; 

20 

.*.  —  =:-the  No.  of  hours  in  going  down  the  river, 

20 

and  ^  ==  "     "  "         "  coming  up         " 

But  the  whole  time  in  going  and  returning  is  10  hours  ; 

Dividing  by  10,  ~ +1=1. 

Multiplying  each  term  of  the  equation  by  3a?, 
2+3=:3:r; 

and  .'.  3.r=5,  miles  per  hour  down, 

20 
/.  the  time  in  going  down  the  river=:— =4  hours,   and   con- 

scquently  the  time  of  returning=10— 4=6  hours. 

6* 


68  ALGEBRA. 

Prob.  21.  A  lady  bought  a  hive  of  bees,  and  found  that  the 
price  came  to  2s,  more  than  f  ths  and  |-th  of  the  price.  Find 
the  price.  Ans,  £2. 

Prob.  22.  A  hare,  50  leap^,  before  a  greyhound,  takes  4 
leaps  for  the  greyhound's  3 ;  btrc  two  of  the  greyhound's  leaps 
are  equal  to  three  of  the  hare's.  How  many  leaps  will  the 
greyhound  take  to  catch  the  hare  ? 

Let  X  be  the  No.  of  leaps  taken  by  the  greyhound  ; 

then  ---  will  be  the  corresponding  number  taken  by  the  hare. 

Let  1  represent  the  space  covered  by  the  hare  in  1  leap ; 

then-  "  "       "  "  "      greyhound" 

.*.  ~  X 1  or  —  will  be  the  whole  space  passed  over  by  the 
o  o 

2       Sx 

hare  before  she  is  taken ;  and  a;  X  -  or  -~  will  be  the  space 

2  ii 

passed  over  in  the  corresponding  time  by  the  greyhound. 
Now,  by  the  problem,  the  difference  between  the  spaces 
respectively  passed  over  by  the  greyhound  and  hare  is  50  X 1, 
or  50  leaps ; 

Sx     Ax 

•'-  2  -  ¥ =^^'        • 

9a;— 8a::=300; 
.-.  ir=300  leaps. 


ON   THE    SOLUTION    OF    SIMPLE    EQUATIONS,    CONTAINING   TWO 
OR    MORE    UNKNOWN    QUANTITIES. 

51.  For  the  solution  of  equations  containing  two  or  more 
unknown  quantities,  as  many  independent  equations  are  re- 
quired as  there  are  unknown-  quantities.  The  two  equations 
necessary  for  the  solution  of  the  case,  when  two  unknown 
quantities  are  concerned,  may  be  expressed  in  the  following 
manner : 

ax-\-hy=:^c 
a'x-\-h'y^^c\ 
Where  a,  6,  c,  a\  6',  c\  represent  known  quantities,  and  x^  y, 


SIMPLE   EQUATIONS.  67 

the  unknown  quantities  whose  values  are  to  be  found  in  terms 
of  these  known  quantities. 

There  are  three  different  methods  by  which  the  value  of 
one  of  the  unknown  quantities  may  be  determined. 

FIRST    METHOD. 

Find  the  value  of  one  of  the  unknown  quantities  in  terms 
of  the  other,  and  the  kno-svn  quantities  by  the  rules  already 
given.  Find  the  value  of  the  same  unknown  quantity  from 
the  second  equation. 

Put  these  two  values  equal  to  each  other ;  and  we  shall 
then  have  a  simple  equation,  containing  only  one  unknown 
quantity,  which  may  be  solved  as  before. 

Ex.  1.      Given  x+yz=z^ -(l))i.i;j  j 

.      ^  x^y=4. U}  to  find  a:  and  3^. 

From  (1)     y=S—x--'-  (a) 

"      (2)     y=x-4: 

Putting  these  twd  values  equal  to  each  other,  we  get 

a;— 4=8— a;, 

2^  =  12; 

.'.x=z6,    < 

By  (a)    y=8-a;=8-6=2.. 

Ex.  2.  Let  x+4y=l6 (1) 

4:x+  y=34 (2) 

From  ^nation  (1),  we  have  x=16—4t/, 

(2)     «     «    rr=?^, 

Hence  by  the  rule,  —j — =16— 4y, 

34— y=64— 16y, 
15^=30; 

.-.  2/ =2. 
It  has  already  been  shown  that  ii;=16 — 4y= (since  y=2; 

and  .-.  4y=8)  16—8=8. 

<i . 

51.  For  the  solution  of  equations  containing  two  or  more  unknown  quan- 
tities, how  many  independent  equations  are  necessary  ?  State  the  ^r«< 
method  of  solution. 


68  ALGEBRA, 

Ex.3. 


5a:+3y=38) ^n.    i^=^ 

-2y=6      f  ^^'-   •|2/=3. 


Ex.4.      2rr-3y=~l 
Sx 


SECOND    METHOD. 

From  either  of  the  equations  find  the  value  of  one  of  tha 
unknown  quantities  in  terms  of  the  other  and  the  known  quan- 
tities, and  for  the  same  unknown  quantity  substitute  this  valu© 
in  the  other  equation,  and  there  will  arise  an  equation  which 
contains  only  one  unknown  quantity.  This  equation  can  be 
solved  by  the  rules  already  laid  down. 

~    Ex.  1.  y-x^^ (1) 

x+y=^ (2) 

From  (1)        y—2+x,        {a) 
This  value  of  y  being  substituted  in  (2),  gives  * 

x+2+x=S, 
2.^=6; 

.\Xz=^,  ^ 

And  by  (a)  y=:2+a:=2+3=5- 
^'^'  ^4?+8.=31  (1) 

Lti_pl0;r  =  192         (2) 

Qeariiig  equation  (1)  of  fractions, 

ir+2+24y=93,  or  a;+24y=:91         (a). 
Gearing  equation  (2)  of  fractions,  ^ 

2/+5+40a;=768,  or  ?/+40a:=763     (/5). 
From  (a)         ir=91— 24?/. 
Substitute  this  value  of  a:,  accordmg  to  the  rule  in  equation 
ifi))  and 

2/+40(91-24y)rr:763, 
or,  y+3640~960y=763; 

.-.  959y=3640-763=:2877, 
and  y=3. 
By  referring  to  equation  (a)  we  have  2;=91—24y= (since 
y =3 ;  and  .-.  24y=72)  91  -72=19. 

Enunciate  ilnQ  second  metTwd  of  solution. 


SIMPLE  EQUATIONS.  69 


4a;+3y=31)  .^     i  x=4 


THIRD   METHOD. 

Multiply  the  first  equation  by  the  coefficient  of  a  in  the 
second  equation,  and  then  multiply  the  second  equation  by  the 
coefficient  of  a;  in  the  first  equation ;  subtract  the  second  of 
these  resulting  equations  from  the  Jirst,  and  there  will  arise  an 
equation  which  contains  only  y  and  known  quantities,  from 
which  the  value  of  y  can  be  determined. 

It  must  be  observed,  however,  that  if  the  terms,  which  in  the 
resulting  equations  are  the  same,  have  unlike  signs,  the  re- 
sulting equations  must  be  added,  instead  of  bemg  subtracted, 
in  order  that  x  may  be  eliminated  (^.  e.,  expelled  from  the 
equations). 

Ex.  1.        Given  5:r+4y=55  -  -  -  (1) 
3a;+2?/=:31  -  -  -  (2) 
To  find  the  values  of  x  and  y 

Mult.  (1)  by  3,  then  15^+12y=165 
"     (2)  by  5,     "    15a;+lQy:^155 

/.  by  subtraction,  we  have   2y=  10        *    . 

.-.  y=     5. 
Now  from  equation  (1)  we  have 

55— 4y 

_55-20 

~"      5 
_35 

■~5 

=7. 

Ex.  2.         Let  the  proposed  equations  be 

ax+h  y=zc    -  -  -  (1) 

a'x+b'x=c' (2) 

Mult.  (1)  by  a',  and  adx-\-a%yz=za^c 
"     (2)  by  a,     "     aa'x+ah'y=.ac' ; 

How  are  equations  solved  by  the  third  metlwd  f 


70  ALGEBRA. 

.•,  by  subtraction,         (a'h — ab  ')y = a'c — ac' 

a'c — ac' 

Mult.  (1)  by  h\  and  ah'x-\-hh'yz=:b  c 
Mult.  (2)  by  b,  and  a'bx-^b'y=^bc' 
By  subtraction,        (ab' —a'b^x-=b'c—bd  \ 

b'c—bc' 

.'.  X- 


Ex.  3.  Let  3a?+4y=29         (1^ 


ab'—a'b' 

'         (1) 
17a;-3y=:36         (2) 


Mtilt.  (1)  by  3,  then  9;r+12y=87 
Mult.  (2)  by  4,  then  68a;--12yr=144. 

The  signs  of  12y  in  the  two  equations  are  unlike ;  .'.  to 
eliminate  y  from  them,  the  two  last  equations  must  be  added 
together ;  and  then 

7707=231 ; 
.-.  ;r=3. 
From  (1)  we  have  4y=29~3^, 

=29—9,  (since  a;=3  ;  and  .-.  3:r=9) 
=20; 


-    Ans,  \  ' 


Ex.4.         Let    6.r4-3y=33)  ^„_    ^  x=S 

;y=5. 

Ex.5.         /<)4a:+3y=31)  ^    ^    ^^^]  J^J 

Ex.6.       r//3^+2y=40)  rAnsA''=l^ 

7 
4. 

IC 

7. 


Ex.7.        rA.\    5^-4y=19)      .    .    .    .    ^^^    <  ^=7 


Ex.8.       /^l   8^+7y=79)      ....    ^^,^  5^=10 


ix= 


Ans. 

=4. 


SIMPLE   EQUATIONS. 


7X 


Ex.  10. 


x+i/ 


-2y=2 


Ex.  11. 


2x—4i/ 

5 
2a:— 3 


67 


2 

5a;— 13?/= 


Ans. 


Arts. 


Ex.  12. 


Sx-'7y=2x+i/-\'l 


-    -    Ans, 


(x=U 

x=S 

y=i 

a;=13 


52.  When  three  unknown  quantities  are  concerned,  the 
most  general  form  under  which  equations  of  this  kind  can  be 
expressed,  is  ax-rby+cz=d         (1) 

a'a;+6V+c'^=^'        (2) 
-    a"a;+6'V+c"2=^"       (3), 
and  the  solution  of  these  equations  may  be  conducted  as  in 
the  following  example : 

Ex.  1.         Let  ^-+%+4.=29     (1)  )   ^^  ^^  ^^^  ^^^^^^ 
3a;+2y+5^=32     (2)   V  of  x  v  z 


I  Multiply  (1)  by  3,  then  6a;+%+12^=87     (4) 
Multiply  (2)  by  2,  then'  6a:+4y+ 10^=64     (5). 

Subtract  (5)  from  (4)  then  5y+  2^=23     (a). 

Multiply  (2)  by  4,  then  12a;+8y+205:=128 
Multiply  (3)  by  3,  then  12a;+92/+  6^=75 

Subtract     ...    -     -2/+14^=53     (/?)• 

II.  Hence  the  given  equations  are  reduced  to, 

5?/+  2^=23     (a) 
-y+14^=53     (/3). 


72  ALGEBRA. 

/ 

Again     -    -     5y+  2z—2S 
Mult.  (i8)  by  5,then-5y+7O0=265 


By  addition     -    -  -  72^=288,  or  ^=^=4 

From  equation  (/3)    '-  -  ^^=14^— 53=56— 53=3. 

trr   T.                ..      /ix  29—3^—4^     29—25 

111.  From  equation  (1)  -       x= ^ — 


2  2 

Ex.2.         x+y+z=90             )  (x=S5 

2a;+40=3y+20      L         -    -  Ans.\y=SO 

2^+40=4^+10      )  (2=25. 

Ex.  3.           x+  y+  z=  6S      )      .  ( :r=24 

a;+2y+32=105      V  Ans.\y=  6 

a:+3y+42=134      )'  (2;  =23. 

PROBLEMS. 

Prob.  1.  There  are  two  numbers,  such,  that  3  times  the 

greater  added  to  ^d  the  less  is  equal  to  36 ;  and  if  twice  the 
greater  be  subtracted  from  6  times  the  less,  and  the  remain- 
der divided  by  8,  the  quotient  will  be  4.  What  are  the 
numbers  ? 

Let  a;=the  greater  number, 
2/= the  less  number; 


Then  3ar+|=36 
6y— 2a?___ 


^^^»  6y-2a;=  32; 


8 

Or,y+9a:=108     (1) 
6y-2a:=  32     (2) 
Mult,  equation  (1)  by  6,  then  6?/+54:r=648 
.Subtract     "       (2)  then  6y—  2x=  32; 

then         56:i;=616; 
..x-^^-U. 

JFrom  equation  (1)       2/=108—9a;= 108—99=9. 

^IProb.  2.  There  is  a  certain  fraction,  such,  that  if  I  add  8 
ttg  the  numerator,  its  value  will  be  ^d ;  and  if  I  subtract  ons 


SIMPLE   EQUATIONS.  73 

from  the  denominator,  its  value  will*  be  |th.     What  is  the 
fraction  ? 

Let  a;=its  numerator,    )  , ,       .i     /.     ^.      -    x 
'    ^  then  the  fraction  is  _. 


i:\' 


y=  denominator,    J  y* 


Add  3  to  the  numerator,  then  — 


y     3 

X         1 


'  5a;=2/— I. 


Subtract  one  from  denom'.,  and        ,  —  ^ 

y— 1     5 

By  transposition^  y—Sx=9     (!) 

y-5r=l     (2). 

'Subjtract  equation  (2)  from  (1),  and  we  have 

2x=S; 

8 
/.  a?=-=4,  the  numerator. 

From  equation  (1)  y=9+3a;=9+12=21,  the  denominator. 

4 

Hence  the  fraction  required  is  — . 

Prob.  3.  A  and  B  have  certain  sums  of  money  ;  says  A 
to  B,  Give  me  £15  of  your  money,  and  I  shall  have  5  times 
as  much  as  you  will  have  left ;  ^ys  B  to  A,  Give  me  £5  of 
your  money,  and  I  shall  have  Exactly  as  much  as  you  will 
have  left.     What  sum  of  money  had  each  ? 

Let  ir=A's  money  )  ,,         4,  i  f^_  i  ^^^^  ^  would  have  after 
y=B's  money  )  (     receiving  £15  from  B. 

♦  y— 15= what  B  would  have  left. 

■    '     -A     '         ,    K       ( what  B  would  have  after 
Again,  y+  5=  <  -  -       n-  e.        \ 

=*      '  ^  '  (      receiving  £d  ii'om  A. 

X—  5= what  A  would  have  left. 

Hence,  by  the  question,  a?+ 15=5 X(y— 15) =5y— 75,  ) 
F  and  y-j-  5=rc— 5. 

Mf  By  transposition,  5?/—  a:=90       (1),) 

f  andy—  x—  —  \0  (2).  f 

Set  down  equation  (1)  5y-—  rr=90. 

Multiply  eq^  (2)  by  5,  5y-5x=-50. 

Subtract  (2)  from  (1)  4a:=140; 

7 


<N^  ALGEBRA. 

140     ^^    ., 
.'.  x=—-=Sd,  As  money. 

From  equation  (1)  5y=90+a;=90+35=125; 

.•.^=-—=25,  B's  money. 
o 

Prob.  4.  What  two  numbers  are  those,  to  one-third  the 
sum  of  which  if  I  add  13,  the  result  shall  be  17  ;  and  if  from 
half  their  difference  I  subtract  one,  the  remainder  shall  be 
two  ?  Ans.  9,  and  3. 

Prob.  5.  There  is  a  certain  fraction,  such,  that  if  I  add  one 
to  its  numerator,  it  becomes  -^ ;  if  3  be  added  to  the  denojni- 
nator,  it  becomes  J.     What  is  the  fraction  1  Ans.  j^. 

Prob.  6.  A  person  was  desirous  of  relieving  a  certain 
number  of  beggars  by  giving  them  2s.  6d.  each,  but  found 
that  he  had  not  money  enough  in  his  pocket  by  3  shillings;  he 
then  gave  them  2  shillings  each,  and  had  four  shillings  to 
spare.  What  money  had  he  in  bis  pocket ;  and  how  many 
beggars  did  he  relieve  1 

Let  a; = money  in  his  pocket  (in  shillings)  ; 
y= number  of  beggars. 

Then  2^Xv  or— =  M^^'  ^^ ^^^'^^^'  which  would  have 
^  2       (      been  given  at  25.  6d.  each, 

and  2Xy,  or  2y=  "         "         at  2s,  each. 


Hence,by  the  question ,-^=rrr+ 3     (1)  \ 

and  2y=x—4     (2). 
Sub*.  (2)  from  (1),  then  |=7,.  or  y=14,  the  No.  of  beggars. 
From  eq".  (2),  a:=22/+4=28+4=32  shillings  in  his  pocket 

Prob.  7.  A  person -has  two  horses,  and  a  saddle  worth 
£10;  if  the  saddle  be  put  on  the  j^rs^  horse,  his  value  be 
comes  double  that  of  the  second;  but  if  the  saddle  be  put  on 
the  second  horse,  his  value  will  not  amount  to  that  of  the 
first  horse  by  £13.     What  is  the  value  of  each  horse? 

Ans,  56,  and  33. 

Prob.  8.  There  is  a  certain  number,  consisting  of  two 
digits.     The  sum  of  those  digits  is  5  ;  and  if  9  be  added  to 


SIMPLE   EQUATIONS.  75 

the  nun^ber  itself,  the  digits  will  be  inverted.     What  is  the 
number  ? 

Here  it  may  be  observed  that  every  number  consisting  of 
two  digits  is  equal  to  10  times  the  left-hand  digit  plus  the 
right-hand  digit :  thus,  34 = 1 0  >>3 + 4. 
Let  x=:left-hand  digit. 
y=iright-hand  digit. 
Th^n  10j;+y=the  number  itself, 
and  10y+a:=:the  number  w^ith  digits  inverted. 
Hence,  by  the  question,  x-{-y=zb  (1), 
and  10a;+y+9z=:10y-fa;,or  9^— 9y=— 9,or a;—y=— 1  (2). 
Subtract  (2)  from  (l),.then  2y=6,  and  2/= 3, 

a;=i5— y=:5  — 3=2; 
.'.  the  number  is  (10.r+2/)=23. 
Add  9  to  this  number,  and  it  becomes  32,  which  is  the 
number  with  the  digits  inverted, 

Prob.  9.  There  are  two  numbers,  such,  that  -J-  the  greater 
added  to  \  the  less  is  13  ;  and  if  -J-  the  less  be  taken  from 
^  the  greater,  the  remainder  is  nothing.  What  are  the  num- 
bers ?  ^715.  18  and  12. 

Prob.  10.  There  is  a  certain  number,  to  the  sum  of  whose 
digits  if  you  add  7,  the'  result  will  be  three  times  the  left- 
hand  digit;  and  if  from  the  number  itself  you  subtract  18, 
the  digits  will  be  inverted,     Wlijfc  is  the  number  1 

Ans.  53. 
Prob.  11.  A  merchant  has  two  kinds  of  tea,  one  worth 
95.  6(f.  per  lb.,  the  other  13s.  6c?.    How  many  pounds  of  each 
must  he  take  to  form  a  chest  of  104  lbs.  which  shall  be  worth 
£56?  .     ^?^s.  33  at  13.S'.  6c?.      . 

71  at    95.  Qd, 
Prob.  12.  A  vessel  containing  120  gallons  is  filled  in   10 
minutes  by  two  spouts  running  successively  ;  the  one  runs  14 
gallons  in  a  minute,  the  other  9  gallons  in  a  minute.     For 
what  time  has  each  spout  run  ? 

Ans,  14  gallon  spout  runs  6  minutes. 
9  gallon  spout  runs  4  minutes. 
Prob.  13.  To  find  three  numbers,  such,  that  the  Jlrst  with 
J  the  sum  of  the  second  and  thii^d  shall  be  120 ;  ^Q^seco7id 
with  \\h  the  difference  of  the  third  and  first  shall  be  70  ;  and 
\  the  sum  of  the  three  numbers  shall  be  ^5. 

Ans.  50,  65,  75. 


76 


ALGEBRA. 


CHAPTER   IV. 

ON  INVOLUTIOI|,AND  EVOLUTION. 


ON  THE  INVOLUTION  OF  NUMBERS  AND  SIMPLE  ALGEBRAIC 
QUANTITIES. 

53.  Involution^  or  "  the  raising  of  a  quantity  to  a  given 
power,"  is  performed  by  the  continued  multiplication  of  that 
quantity  into  itself  till  the  number  of  factors  amounts  to  the 
number  of  units  in  the  index  of  that  given  power.  Thus,  the 
square  of  a=aXci^=cb^  \  the  cube  of  b=bX^Xl>=b^ ',  the 
fourth  poiver  of  2=2x2x2x2=16;  the  Jiftk  poiver  of  3 

=3x3x3x3x3=243;  &;c.,  &;c. 

54.  The  operation  is  performed  in  the  same  manner  for 
simple  algebraic  quantities,  except  that  in  this  case  it  must  be 
observed,  that  the  powers  of  negative  quantities  are  alter- 
nately +  and  —  ;  the  ev^/i  powers  being  positive,  and  the  odd 
powers  negative.  Thus  the  square  of  +2a  is  +2aX  +2a  or 
+4a^;  the  square  of  —2a  is  —  2aX  —  2a  or  +4a^;  but  the 
cube  of  — 2a  =  --2aX— 2aX— 2a=+4a2x— 2a=— 8al 


The  several  powers  of  - 
are,  ^ 

a     a     Oj^ 

a     a     a_a^ 

Cube   =lXlX^-^' 

4th  _  a     a     a     a     a^ 

p,^er--X^X^X^=P 

&;c.=&c. 


Mi( 


d  the  several  powers  of— ;r-, 
2c 


Squ. 


~     2c^     2c~"'"4c^' 


4th 
power  - 


2c 
h 


'2'c 
b 


X 


2c 


2c 


X- 


b_ 

b 


2c 


X- 


"Sc^' 
b^ 

"2c 


=  +  16?' ^^•^^^- 


ON  THE  INVOLUTION  OF  COMPOUND  ALGEBRAIC  QUANTITIES. 

55.  The  powers    of  compound   algebraic   quantities   are 

53.  What  is  involuti(5^?  How  is  it  performed? — 54.  In  what  manner  ia 
invohiti  on  performed  for  simple  algebraic  quantities? — 55.  How  are- the 
powers  of  compound  quantiti  3S  raised  ? 


INVOLUTION.  77 

raised  by  the  mere  application  of  the  Rule  for  Compound 
Multiplication  (Art.  34).     Thus, 

Ex.  1.  What  is  the  square  Ex.  2.  What  is  the  cube  of 

ofa+26?  a'-xl 

a +26                        ^  a^-x 

a  4-26  a^^x 


Square  =a^+4a6+ 46^  Square = a*— 2a^a;-fiu^ 


_-^_a^+2a'^^--^ 
Cube=a«-3a^ar+3aV-a;8 


Ex.  3. 

Whft  is  the  5th  power  of  a+6 1 
a +6 

a+5 

+  ah  +6^  ^^ 

a^  +  2a6 + 6^ = Square 

' ___^  « 

a^+2a^6+   ay 

+    a^6+  2«^>'  +6^ 

aH  3a^6+  3a6^ +63=Cube 
a+       6 

a*+3a^6+  3a'6^+    a6^ 
+    a^6+  3a^6^+    3a6^+6^ 

^*+4^^+  6a'Z»'+    4a5^+^>'*=4th  Power 
a_jr_b 

a^J^^a'h  +   ^aW-\-  Wh''-\-ah^ 
+  a^6  +  4a^6^+  6a^6^+4a5^+6^ 

gs^  5a46  +  10ag6^  +  10a^6^+5a6'+6^=:5th  Power, 
7* 


78  *         ALGEBRA. 

Ex.  4.  The  4:'^  poiver  of  a+Sb  is  a*+12a^6+54a^Z>'^+108a5» 
+  816^ 

Ex.5.  The  square  of  Sx'+2x+5  is  9x'+12x^+S4.x^+2(h 
+25. 

Ex.  6.  The  cw6e  of  Sx-5  is  W-135a;^+225a;--125. 

Ex.  7.  The  cube  of  a;^-2a:+l  is  ic«-6aj^+15^*-20;r^+15a?« 
-6x+l. 

'  Ex.  8.  The  square  of  a+5+c  is  a2+2a5+62+2ac+25c+d 

ON    THE    EVOLUTION    OF    ALGEBRAIC    QUANTITIES. 

56.  Evolution^  "  or  the  rule  for  extracting  the  root  of  any 
quantity,"  is  just  the  reverse  of  Involution  ;  and  to  perform 
the  operation,  we  must  inquire  what  quantity  multiplied  into 
itself,  till  the  number  of  factors  amount  to  the  number  of 
units  in  the  index  of  the  given  root,  will  generate  the  quantity 
whose  root  is  to  be  extracted.     Thus,  % 

(1.)    49=7 X7 ;  .-.  the 5^. roo^of  49  (or by  Def^  15,y'49)=:7. 

m  i^^j-b^^-bX-bX-b)  ,\cube  rooto^-¥  {^Z:h')  =  '-'^' 
•^    _     16a*_2a     2a     2a    2a    ^    4   /16a*_2a 

^  '^  8iJ*~36  ^36^36  ^S^"*'*  V  8l6"*~'36' 

(4.)     32=2X2X2X2X2;. -.^32=2, 

(5.)       a'^^a^ Xa'Xa^',  /. ^a'-a\ 

Hence  it  may  be  inferred,  that  any  root  of  a  simple  quart,* 
tity  can  be  extracted^  by  dividing^^  its  index,  if  possible,  by  the 
index  of  the  root, 

57.  If  the  quantity  under  the  radical  sign  does  not  admit 
of  resolution  into  the  number  of  factors  indicated  by  that 
sign,  or,  in  other  words,  if  it  be  not  a  complete  power,  then  its 
exact  root  cannot  be  extracted,  and  the  quantity  itself,  with 

.     the  radical  sign  annexed,  is  called  a  Surd.      Thus  'y/37,  J/a\ 
V/>^,  ^47,-  &c.,  &c.,  are  Surd  quantities. 

. » 

56.  What  is  EvolMtion  ?  How  is  ifc  performed  ?— 57.  What  is  a  Surd  quantity  f 


EVOLUTION.  79 

58.  In  the  involution  of  negative  quantities,  it  was  observed 
that  the  even  powers  were  all  +,  and  the  odd  powers  —  ; 
there  is  consequently  no  quantity  which,  multiplied  into  itself 
in  such  manner  that  the  number  of  factors  shall  be  even^  can 
generate  a  negative  quantity.     •Hence  quantities  of  the  form 

-/^^  -^=ao~y^^  V-^^  -V^^^  '^^•'  ^^'^  ^^^^  ^^ 
real  root,  and  are  therefore  called  impossible, 

59.  In  extracting  the  roots  of  compound  quantities,  we 
must  observe  in  what  manner  the  terms  of  the  root  may  be 
derived  from  those  of  the  power.  For  instance  (by  Art.  55, 
Ex.  3),  the  square  of  a+5  is  a^-f-2a/^+^^  where'the  terms  are 
arranged  according  to  the  powers  of  a.  On  comparing  a-^b 
with  a^-|-2a6+Z>^,  we  observe  that  the  first  term  of  the  power 
(a^)  is  the  square  of  the  first  term  of  the 

root  {a).     Put  a  therefore  for  the  first  o}-\-2ah-\-b^  la-^-b 
term  of  the  root,  square  it,  and  subtract  o?  \ 

that  square  from  the  first  term  of  the  %^  hA.}fi 

power.     Bring  down  the  other  two  terms  2^+^  Io^7,Ia2 
2ah-\-'t\  and  double  the  first  term  of  the  |2aH-6_ 

root;  set  down  2a,  and  having  divided  ^ ^ 

the  first  term  of  the  remainder  (2a6)  by  " 

it,  it  gives  5,  the  other  term  of  the  root ; 

and  since  2ab-\'b'^=^{2a-\-b)b^  if  to  2a  the  term  b  is  added, 

and  this  sum  multiplied  by  5,  the  result  is  2a5-j-6'^;  which 

being  subtracted  from  the  two  terms  brought  down,  nothing 

remains. 

60.  Again,  the  square  of  a+^  +  c  (Art.  55,  Ex.  8.)  is  a^+I 
2ab-\-b'^-{-2ac-{-2bc-\-c^\    in    this    case    the    root    may    be? 

continuingthe     2a+i|2«J+6« 
process  m  the  L  z,     72 

last     Article.  \}±^ 


2ac+2bc+c'' 
2ac-\-2bc—c^ 


Thus,  having  2a+2b-{-c 

tound  the  two 

first  terms  {a-\-b)  *       %       * 

of  the  root  as  '"^ 

before,  we  bring  down  the  remaining  three  terms  2ac-\-2bc 

58.  Explain  the  nature  of  an  impossible  quantity. — 59.  How  are  the  roots 
of  compound  quantities  extracted  ? 


80  ALGEBRA. 

+  c*  of  the  power,  and  dividing  2ac  by  2a,  it  gives  c,  the 
third  term  of  the  root.  Next,  let  the  last  term  (J)  of  the 
preceding  divisor  be  doubled,  and  add  c  to  the  divisor  thus 
increased,  and  it  becomes  2a+26-|-c;  multiply  this  new 
divisor  by  f,  and  it  gives  2ac-t-26c+c^,  which  being  subtracted 
from  the  three  terms  last  brought  down,  leaves  no  remainder. 
In  this  manner  the  following  Examples  are  solved.^ 

Ex.  1. 


^x' 


4:X'  + 


6x'+^x' 
4 


A  8  .  Q    IK  \     20aj^+15a:+25 
4:X'\-Sx-i-i)  j     20^;^ 4- 15^^+25 


Ex.2. 

x^+4:x'+2x^+9x^-4x+4:{x^+2x^-^x+2 


2x^+2x')4:x'+2x* 
4x'+4.x' 

2x^+4.x^-x)-2x^-{-9x''-4x 
-2x^-4x^+  x'' 

2x^+4x'^2^A^+4:x'+Sx'-4x+4t 
+4aj'+8a;'— 4^+4 


Ex.  3.  The  square  root  of  4x^+4x7/+y''  is  2^+y. 

Ex.  4.  The  square  root  of  25a^+30a5+95'  is  5a +85. 

Ex.  5.  Find  the  square  root  of  9a;*+12^^+22a?^+12;r+9. 

Alls.  Sx'-{-2x+S. 


EVOLUTION.  81 


Ex.  6.  Extract  the  square  root  oMx^—Wx^+24:X^-'l6x+4. 

Ans.  2x'—4x-y2. 

4 
"9 


Ex.  7.  Find  the  square  root  of  S6x'~S6x'+l7x'-4x'^^' 


'  2 

A71S.  60;"^— 3.1; -f-. 
o 

Of)        1  /» 

Ex.  8.  Extract  the  square  root  of  a;^+8^'+24-| — --\ — 5. 

X  X 

4 

•4^5.  a;^+4H — 5. 


ON  THE  INVESTIGATION  OF  THE  RULE  FOR  THE  EXTRACTION 
OF  THE  SQUARE  ROOT  OF  NUMBERS. 

Before  we  proceed  to  the  investigation  of  this  Rule,  it  will 
be  necessary  to  explain  the  nature  of  the  common  arithmeti- 
cal notation. 

61.  It  is  very  well  known  that  the  value  of  the  figures  in 
the  common  arithmetical  scale  increases  in  a  tenfold  propor- 
tion from  the  right  to  the  left ;  a  number,  therefore,  may  be 
expressed  by  the  addition  of  the  iinits^  tens^  hundreds^  &c.,  of 
which  it  consists.  Thus  the  number  4371  may  be  expressed 
m  the  following  manner,  viz.,  4000  +  300+70  +  1,  or  by  4X 
1000+3x100+7x10  +  1 ;  hence,  if  the  digits*  of  a  number 
be  represented  by  a,  ^,  c,  d^  e,  &c.,  beginning  from  the  left 
hand,  then, 

A  No.  of  2  figures  may  be  expressed  by  10a +5. 
.     "      3  figures  "  by     lOOa+105  +  c. 

"      4  figures  "  by  1000a  + 1006 +  10c+cf. 

&;c.  &c.     &c. 

62.  Let  a  number  of  three  figures  (viz.,  lOOa+105+c)  be 

*  By  the  digits  of  a  number  are  meant  the  figures  which  compose 
it,  considered  independently  of  the  value  which  they  possess  in  the  arith- 
metical scale.  Thus  the  digits  of  the  number  537  are  simply  the  num- 
bers 5,  3,  and  7  ;  whereas  the  5,  considered  with  respect  to  its  place  in 
the  numeration  scale,  means  500,  and  the  3  means  30. 


61.  Explain  the  common  arithmetical  scale  of  notation.  What  is  a 
digit? — 62.  Show  the  relation  between  the  algebraical  and  numerical 
method  of  extracting  the  square  root,  and  that  they- are  identical. 


82  ALGEBRA. 

squared,  and  its  root  extracted  according  to*  the  Rule  in  Art. 
60,  and  the  operation  will  stand  thus ; 

I.  10000a'+2000ah  +  100b'+200ac+20bc+c\100a+l0b+c 
IQQOOa^  ^ 

200a +  106)  2000a5  + 1006^ 
2000a6+100^>^ 


200a+206+c)  200aH-205c+e* 
200ac+20bc-\-c' 


,  _o  /  and  the  operation  is  transformed  into  the  fol- 
^ZiC      lowing  one; 

40000+12000+900+400+60+lf200+30+l 
40000  V 

400 + 30^ 1 2000 + 900 + 400 
712000+900 

4OO+6O+1W0+6O+I 
7400  +  60+ 1 


III.  But  it  is  evident   that  this   operation  would  not  be 
affected  by  collecting  the  several   numbers  which  stand  in 
the   same   line    into   one    sum,   and 
leaving    out   the    ciphers   which   are  ^qqai/oqi 

to  be  subtracted  in  the  several  parts  oddblf  2dl 

of  the  operation.     Let  this  be  done  ; 

and  let  two  figures  be  brought  down  431133 
at  a   time,  after   the  square   of  the  1129 

first  figure  in  the  root  has  been  sub-  461 
tracted;  then  the  operation  may  be 
exhibited   in   the    manner    amiexed;  : 

from  which  it  appears  that  the  square  ::::;::= 
root  of  53,361  is  231.  ' 


461 
461 


QUADRATIC  EQUATIONS.  83 

63.  To  explain  the  division  of  the  given  r  umber  into 
periods  consisting  of  two  figures  each,  by  placing  a  dot  over 
every  second  figure  beginning  with  the  units  (as  exhibited  in 
the  foregoing  operation),  it  must  be  observed,  that,  since  the 
square  root  of  100  is  10 ;  of  n),000  is  100  ;  of  1,000,000  is 
1000,  (fee,  &c. ;  it  follows  tharthe  square  root  of  a  number 
less  than  100  must  consist  of  one  figure ;  of  a  number  be- 
tween 100  ancif  10,000,  of  two  figures,  of  a  number  between 
10,000  and  1,000,000,  of  three  figures,  &;c.,  &c. ;  and  conse- 
quently the  number  of  these  dots  will  show  the  number  of 
figures  contained  in  the  square  root  of  the  given  number. 
Thus  in  the  case  of  53361  the  square  foot  is  a  number  con- 
sisting of  three  figures. 

Ex.  1.  Find  the  square  root  of  105,625.  Ans.  325. 

Ex.  2.  Find  the  square  root  of  173,056.  Ans,  416. 

Ex.  3.  Find  the  square  root  of  5,934,096.         Ans,  2436. 


CHAPTEE    V. 


ON  QUADRATIC  EQUATIONS. 


64.  Quadratic  Equations  are  divided  into  pure  and  adfected. 
Pure  quadratic  equations  are  those  which  contain  only  the 
square  of  the  unknown  quantity,  such  as  a:'^36;  a;^+5= 
54;  ax^—h:=c\  &c.  ^^ec^eo?  quadratic  equations  are  those 
which  involve  both  the  square  and  simple  power  of  the  un- 
known quantity,  such  as  x^+4x=46 ;  3a;^— 2ar=21 ;  ax'^-{- 
''2bx  =  c-\-d',  &c.,  &c. 


63.  Explain  the  principle  of  the  rule  and  the  object  of  pointing  off  in 
extracting  the  square  root  of  numbers. — 64.  How  are  quadratic  equations 
divided  ?     What  is  an  adfected  quadratic  equation  ? 


\ 


84  ALGEBRA. 

ON  The  solution  of  pure  quadratic  equations. 

65.  The  Rule  for  the  solution  of  pure  quadratic  equations 
is  this :  "  Transpose  the  terms  of  the  equation  in  such  a  man- 
ner, that  those  which  contain  x^  may  be  on  one  side  of  the 
equation,  and  the  known  qud^iities  on  the  other ;  divide  (if 
necessary)  by  the  coefficient  of  x^ ;  then  extract  the  square 
root  of  each  side  of  the  equation,  and  it  will  give  the  values 
of  x:' 

Ex.  1. 

Leta;^+5=54. 
By  traiRposition,  0^^=54—5=49. 
Extract  the  square  root ) 

of  both  sides  of  the  [  then  x=±^4:9=:±7. 
equation,  ) 

Ex.2. 
Let  Sx'-4.=zlfh 

By  transposition,  3.c^=71 +4=75. 

75  '  ' 

Divideby3,  .1:^=— =25. 

Extract  the  square  root,  a;=  +  ^/25=="l"5. 
Ex.3. 
Let  ax^—b=:c; 
then  ax'^=zc-\-bj 

and  x'^= 

V      a 
Ex.  4.       5a;*— 1       =244  -    -    Ans.  x=+7. 

Ex.  5.       9^^+9       =3a;^+63      Ans.  x=±S. 

Ex.  6,       — ^       =45     -    .     Ans.  x=  +  10, 
Ex.  7.       bx'+c+S  =2bx'+l      Ans.  x=  +  ^'±3, 
65.  State  the  rule  for  solving  pure  quadratic  equations. 


QUADRATIC   EQUATIONS. 


85 


ON  THE  SOLUTION  OF  ADFECTED  QUADRATIC  EQUATIONS. 

66.  The   most   general   form    under   which   an   adfeeted 

quadratic  equation  can  be  exhibited  is  ax^-{-bx=:c',  where 

a,  6,  c  may  be  any  quantities  whatever,  positive  or  negative, 

integral  or  fractional,     Dividff  each  side  of  this  equation  by 

be  be 

a,  then  x^+-x=-»      Let  -=p,  -—q\  then  this  equation  is 
a        a  a         a 

reduced  to  the  form  x^+px=^q,  where  p  and  q  may  be  any 
quantities  whatever,  positive  or  negative,  integral  or  frac- 
tional. 

67.  From  the  twofold  form  under  which  adfeeted  quad- 
ratic equations  may  be  expressed,  there  arise  two  Rules  for 
their  solution. 


EULE  I. 

Let  x'^'^  pxr=zq. 
Add |- to  each  side)    ,4,       ^^^i>^       ^^^+4^ 


of  the  equation,  then 

Extract  the  square  root 
of  each  side  of  the 
equation,  then 


x'X.P^+r 


^+r- 


—2-  2 

and.=.±^±±^±^. 


Hence  it  appears,  that  "  if  to  each  side  of  the  equation 
there  be  added  the  square  of  half  the  coefficient  of  x,  there 
will  arise,  on  the  left-hand  side  of  the  equation,  a  quantity 
which  is  a  complete  sqimre  ;  and  by  extracting  the  square  root 
of  each  side  of  the  resulting  equation,  we  obtain  a  simph 
equation,  from  which  the  value  of  x  may  be  determined.'' 

*  Since  the  square  of  -{-a  is  -\-a\  and  of  — a  is  also  +a*,  the 
square  root  of  ■\-aP'  may  be  either  -\-a  or — a;  henc»^e  square  root 
of  p2.^4^  may  be  expressed  by  X  -y/p^-j-^^'* 


66.  What  is  the  most  general  form  of  a  quadratic  -equation  ?    Can  it  be 
reduced  to  another  form  ? — 67.  Enunciate  the  1st  Rule. 


86  ALGEBRA. 

68.  From  the  form  in  which  the  value  of  x  is  exhibited  in 
each  of  these  Rules,  it  is  evident  that  it  will  have  tivo  values ; 
one  corresponding  to  the  sign  +,  and  the  other  to  the  sign—, 
of  the  radical  quantity. 

El4  1. 

Let  ir'+8a:=65. 
Add  the  square  of  4  (i.  e,  16)  to  each  side  of  the  equation, 
then       -     -    -     a;^+8a:+16=:65  +  16=:81.      . 
Extract  the  square  root  of  each  side  of  the  equation,  then 

ir+4=+v/81  =  +9,   -► 
and  a;  =  9  —  4  =  5 ; 

OVX  =— 9— 4:zi— 13. 

Ex.2. 

Let  a;*— 4a; =45. 

Add  the  square  of  )  ^'i_a^a.a-ak^a-ao 
2  (^.  e.  4),  then  [  ^      4a;+4_4D+4_4y. 

Extract  the  square  root,  and  a;— 2=  +y^49— +7, , 
andrr=7+2=:9; 
or,  a;=:2— 7=  — 5. 
Ex.3.         x''+\2x=im   -    -    -    -     Ans.     x=z  6  or —18. 
Ex.4.         x'^—\4:X=i  51   -    -    -     -    Ans,     ir=17  or  —  3. 
Ex.5.         x^—  6ar=  40  -    -     -     -     Arts,     rr=10  or  —  4,  s 

Ex.  6.  a:^— 5a:=6. 

In  this  example  the  coefficient  of  x  is  5,  an  odd  number. 

5 

Its   half  is  -;  and  .*.   adding  to  each  side  of  the  equation 


/5\*      25 


we  get 


^  ,  25    24+25    49 

zQ  +  -—= =— -. 


O  I  7 

Extracting  the  square  root,  x—-  =_^  > 


^4-7     ^ 


•       QUAI>KATIO  EQUATIONS.  87 

Ex.7.  x'^x^zQ, 

Here  the  coefficient  of  a:  is  1 ;  adding  therefore  {^Y  or  ^ 
to  both  sides,  we  get  / 

Extracting  the  square  root,  x—^='^~ ; 

/.  a;r=^+-=:3or— 2. 

<w 

Ex.    8.         a;^+7a;=78. ^?i5.  x=6  or  —13. 

Ex.    9.         x^+Sx=2S. Ans.  x—4:  or  —  7. 

Ex;  10.         a;^— 3a:=40. Ans.  a;=8  or  —  5. 

Ex.  11.         x^+  ir=30. Ans,  a;=5  or  —  6. 

Ex.12.  Let7a;2-20^=32;  fin^rr. 

Dividing    by    7,  ic^— -y^=y. 

CompF.)    3^20       /10y32     100     224  *  100     324 

the  sq.  P  ~  7  ^+\  7  j  -  7  +  49  -  49  +  49  -  49  • 

Hence,  x-—=±^^-^=:±—', 

.        10,  18      .  ^,     . 

and  ir=— -lL--=4  or  —If. 
7       7  ^ 

Ex.  13.  5^^+4a:=z273. 

Dividing  by  5,  a;^+-.r=— — . 

To  each )     /2\=      4       ^,4,4     273  ,  4      1369 
side  add     \    (5J°^25""'^" +5^+25=^+25=^5- 

2         37 

Extracting  the  square  root,  a;+r  — "I"— » 

o  o 

_37_2 
Ex.14.         3a;H2:r=161    -    -    -    -    ^^5.  a;  =  7  or -7|. 


,  ^=+^_^^7,  or  -71. 


88  ALGEBRA. 

Ex.15.  2x^—5x=lllf       -     .    -  Ans.  x=  9  or  —   ^^, 

Ex.16.  So;^— 2a:=:280       -     -     -  Ans.  x =10  or  -  9^, 

Ex.17.  4x'-7x=:492y  '    -    -  Ans.  xz=zl2  or -lOi. 


A  quadratic  equation  seldom  appears  in  a.  form  so  simple 
as  those  of  the  preceding  examples  ;  it  is  therefore  generally 
found  necessary  to  employ  in  the  solution  of  a  quadratic  the 
following  reductions. 

(1.)  Clear  the  equation  of  fractions. 

(2.)  Transpose  the  terms  involving  x'^  and  x  to  the  left- 
hand,  and  the  numbers  to  the  right-hand  side  of  the  equation. 

(3.)  Divide  all  the  terms  of  the  equation  by  the  coefficient 
of  x'^, 

(4.)  Complete  the  square. 

(5.)  Extract  the  square  root  of  both  sides,  and  there  will 
arise  a  simple  equation,  from  which  the  value  of  x  may  be 
found.        ^ 

Ex.1.  3--"4- 

Multiply  by  3,  and  4^^—33=^. 
By  transposition,       4^;^—  a; =33. 

^.  ..      .       .         ,       ,     1       33 

Divide    by  4,   and    x^—-:X=-—: 
J      •>  4        4 

Complete  the   )     ^_\        1  _33      1  _528      1  _529 

square,        j  ^      4^"^64~  4  ''"64""  64  "^64~  64 

i 
1         23 
Extracting  the  sq.  root,  x—-=^—' 
o  o 

.,.4+1^3  or  -21. 

Q         4- 

Ex.  2.  _^+*=5. 

4a;+4 
9-) =5a;+5. 


QUADRATIC   EQUATIONS.  89 

6x^-Sx=4, 

2     8    ,  16_4     16_36 

4      ,6 

and  x  =  -J2.-z=2  or  — --. 
5     5  5 

Ex.3.       —-Izzrar+ll.      ...     -     ^715.     a:=:12or-a 
o 

xLx.  4.       -^+-=0 Ans.     a;=:3  or  L 

3      ic     3  ^ 

Jix.  5.       -——-=9. Ans,     ar=6  or  — -. 

Ex.  6.       -4t+?=3     ....    -     ^7i.s.     a;=2  or  -1 

Ex.  7.       a;'— 34=^0;. ^ns.     a;=6  or  — 5|. 

Ex.  8.       f +- =51. ^/i5.     ir=25  or  1. 

5     .T 

24 

Ex.9.       a;H -=3a;-4.      -    -    -    Ans.     a;=5  or  — 2, 

x  —  \ 

Ex.  10.     -4_+^±l=^.      -    .     .     ^n^.     x=2  or  -3. 
a;+l        ^         ^ 

Ex.11.     — -—=.^—9.-     -     -     Ans.    a;=10  or  — — -. 

a;+2        6  7 

Ex.  12.  Given  x^+^x=-U  ;  find  x. 

ic^H-8a?+16=16-31  =  -15, 
0:4-4  =  +  ^"^^^; 
.%  a;=  ~4+'v/-15,  &  a.—  -4  -  'v/-15, 

both  of  which  are  impa  ssihle  or  imaginary  values  of  x. 


90  ALGEBRA. 


Ex,  13.       x"—  2x=—  2.      -.  .     -     Ans.     a;=l  +  V~l. 
Ex.14.       a;^— 16.^=— 15.       -    -     -    Ans,     ir=:15  or  1. 
Ex.  15.  Let  lSx'+2x=60. 

Divide  by  13,  ^^+?^=g. 


Add  the       1         2^       1   _60       1   __780       1   _781 
square  of  J3  p +13+169~13"^169~"169"^169~169* 
Extract  the  I         1  __|_y781_  ^27.94 
square  root  f  ^  ""is""" —    13    ~~ —    ^3 

+27.94-1     26.94     ^  ^^  ^  ..n^ 

/.  x==^ — =:— — -=2.07  or  —2.226. 

13  lo 

Ex.  16.     rr^-6^+19==13.        -     Ans.     a;=4.732  or  1.268. 

Ex.  17.     5^^+4:r=25.         -    -     Ans.     ic=1.871. 

Any  equation,  in  which  the  unknown  quantity  is  found  only 
in  two  terms,  with  the  index  of  the  higher  power  double 
that  of  the  lower,  may  be  solved  as  a  quadratic  by  the  pre- 
ceding  rules. 

Ex.  18.  Let  ir«-2aj^=48. 

Complete  the  square,  x^—2x^-{- 1  =49 ; 
Extract  the  square  root,    .'T^— l  =  +  7; 

.-.  ^^3=8  or  — 6; 
and  .-.  X  =2  or^__^^ 
Ex.  19.  2a;-7-y/a7=99. 

7    ,      99 

^--2^-"=¥' 
7  ,  ,  ny   99  ,  49   841 

.      7  ,29     .  11 

V^=j±-j=9or--. 

121 

•.  by  squaring  both  sides,  ^=81  or  — — . 


QUADRATIC   EQUATIONS.  91 

Ex.20.       x^-\-4x''=l2,   -    -    Ans.     x=  +  ^2oT±jy/^^. 
Ex.  21.       a;«-8^«=513.      -     A71S.     x=S  or^^^^oT 

Rui#  II. 

Let  ax^^bx=Cy 
Multiply  each  side  of  )  ^j^^^  4„v+4a6^=.4ac. 

the  equation  by  4a,    j  — 

Add  6^  to  each  side,  1  4^,^,+  ^^^^     6^=4ac+6^ 
we  nave  \  — 


Extract  the  square  root  as  before,  2aa;_+5  ==  + V4ac+6^ 


.-.  2aa;=  +  'v/4ac+62q:5. 

and.=±V;.5!±?±*. 
2a 

From  which  we  infer,  that  "  if  each  side  of  the  equation 
be  multiplied  by  four  times  the  coefficient  of  .r^,  and  to  each 
side  there  be  added  the  square  of  the  coefficient  of  x,  the 
quantity  on  the  left-hand  side  of  the  equation  will  be  the 
square  of  2ax~^b.  Extract  the  square  root  of  each  side  of 
the  equation,  and  there  arises  a  simple  equation,  from  which 
the  value  of  x  may  be  determined."* 

If  a=l,  the  equation  is  reduced  to  the  form  x^^px=q', 
in  this  case,  therefore,  the  Rule  may  be  applied,  by  "  multi- 
plying each  side  of  the  equation  by  4,  and  adding  the  square 
of  the  coefficient  of  a;." 

From  the  form  in  which  the  value  of  x  is  exhibited  in  this 
Rule,  it  is  evident  that  it  will  have  two  values ;  one  corre- 
sponding to  the  sign  +,  and  the  other  to  the  sign  — ,  of  the 
radical  quantity. 

Ex.  1. 

Let3a;^+5a:=42. 
Multiply  each  side  of  the  ) 

equation  by  (4a)  12 ;  [•  36a:'+60a;=504. 
then  ) 

*  The  principle  of  this  Kule  will  be  found  in  the  Bija  Ganita^  a 
Hindoo  Treatise  on  the  Elements  of  Algebra.  For  a  full  account  of  that 
work,  as  translated  by  Mr.  Strachey,  see  Dr.  Hutton's  Tracts^  vol.  IL 
Tract  33. 


92  ALGEBRA. 


Add  (h^)  25  to  each  side  }  oa  i  t  an    i  ok     ka^  .  oc     cor. 
of  the  equation,  we  have  }  36^^+60^+^5=504+25=529. 

Extract  the  square  root  of  each  side  of  the  equation,  which 

gives 6a;+5=+V529=+23; 

.-.^^=+23-5=18  or  -28; 


and  rr=— -=3, 

D 

28         14 


Ex.  2. 

Let  a;*— 15a:=— 54. 
Multiply  by  4,  then  4x^—60x=—216, 

^lich  s?de    \  ^^^  4:r«-60a:+225=225~216=9. 
Extract  the  square  root,  2ar---15=+ v/9=+3; 

.-.  2a:=15+3=18or  12, 

18      12    ^       ^ 
and  x=—  or  -77-= 9  or  6. 


ON  THE  SOLUTION  OF  PROBLEMS  PRODUCING  QUADRATIC 
EQUATIONS. 

69.  In  the  solution  of  problems  which  involve  quadratic 
equations,  sometimes  both,  and  sometimes  only  one  of  the 
values  of  the  unknown  quantity,  will  answer  the  conditions 
required.  This  is  a  circumstance  which  may  always  be  very 
readily  determined  by  the  nature  of  the  problem  itself. 

Problem  1. 

To  divide  the  number  56  into  two  such  parts,  that  their 
product  shall  be  640. 

Let  x-=one  part, 
then  56 — ii:=the  other  part, 
and  X  (56— rr)=  product  of  the  two  parts. 
Hence,  by  the  question,  x  (56— a;) =640, 
'  or  56a:-a:^=:640. 


QUADRATIC  EQUATION'S.  93 

By  transposition,  x^—56x—  -640. 

By  completing  the  square, )  ^«_56;,+784=784-640=144  • 
(Rule  I.)  j 

.\  a;~28  =  + V144=±12, 
and  ^=28+12=40  or  16. 
In  this  case  it  appears  that  the  two  values  of  the  unknown 
quantity  are  the  two  parts  into  which  the  given  number  was 
required  to  be  divided. 

Prob.  2.  There  are  two  numbers  whose  difference  is  7,  and 
half  their  product  plus  30  is  equal  to  the  square  of  the  less 
number.     What  are  the  numbers  1        ^ 

Let  a;=the  less  number, 
then  .r-}-7=the  greater  number, 

and ^r ^+30— half  their  product  ^^W5  30. 

Hence,  by  the  question, ^ — ^ + 30 =a;'^  (square  of  less)^ 

Multiply  by  2     -    -     x'-\-lx-{-m-1x\ 
By  transposition      -  o;^— 7a:=:60. 

™9  ^rJle  no  ""^^  \  4^'~28.:+49=240+49=289, 
.-.  2a;-7=y289=17 

2a;=17+7=24,  or  x^l2  less  number; 
hence  ir+7=:12-|-7=19  greater  number. 

Prob.  3.  To  divide  the  number  30  into  two  such  parts, 
that  their  product  may  be  equal  to  eight  times  their  dit 
ference. 

Let  a;=the  less  part, 
then  30— -a: = the  greater  part, 
and  30— or— a;  or  30— 2^:=  their  difference. 
Hence,  by  the  question,  .tX(30— a;)=8x(30— 2.r), 
or30.r-ir^=:240-16ar. 


94  ALGEBRA. 

By  transposition  ,x'^—4:6x=-^  240. 
^■^"^^(W^^^  }  ^^-46^+52i=529-240=280; 

V-  ^'^-23=±y'289=:±17, 
,  and  a;=''23^117=40  or  6=less  part ; 

30 —^=30—  Qr=:24:=:greafer  part. 

In  this  case,  the  solution  of  the  equation  gives  40  and 
for  the  less  part.     Now  as  40  cannot  possibly  be  a  part  of 
80,  we  take  6  for  the  less  part,  which  gives  24  for  the  greater 
part ;  and  the  two  numbers,  24  and  6,  answer  the  conditions 
required. 

Prob.  4.  A  person  bought  cloth  for  £33  155.,  which  he 
sold  again  at  £2  8s.,  per  piece,  and  gained  by  the  bargain 
as  much  as  one  piece  cost  him.  Required  the  number  of 
pieces. 

Let  rr=the  number  of  pieces. 

675 

Then =the  number  of  shillings  each  piece  cost, 

and  48:r=the  number  of  shillings  he  sold  the  whole  for; 
.*.  48aj— 675= what  he  gained  by  the  bargain. 

675 

Hence,  by  the  problem,  48a;— 675= . 

By  transposition  )     ^     ^^^  _225 
and  division,    |  ^       16"^ "16"* 

Complete  the )     ,__225      50625 _225     50625_65025 
sq.  (Rule  I.)  J  ^      Jq^'^  1024  "'  16  "^1024  ~  1024* 

225         /65025     255 
.'.  X — 


V 


32  ~V    1024"  32' 

255+225     ,, 
and  x= ~- =15. 

Prob.  5.  A  and  B  set  off  at  the  same  time  to  a  place  at 
the  distance  of  150  miles.  A  travels  3  miles  an  hour  faster 
than  B,  and  arrives  at  his  journey's  end  8  hours  and  20 
minutes  before  him.  At  what  rate  did  each  person  travel 
per  hour  ? 


QUADRATIC  EQUATIONS.  95 

Let  rr=rate  per  hour  at  which  B  travels. 
Theiia;+3=:  "  "  A       " 

150 

And = number  of  hours  for  which  B  travels. 

X 


ir+3 

But  A  is  8  hours  20  minutes  (8^  hours)  sooner  at  his  jour« 
ney's  end  than  B ; 

150   ,oT_150 


Hence  ^  ,  ^+8|-=  — ^, 


150      25_150 

By  reduction,  ic^4-3a;=54. 

9  9     225 

Complete  the  square,  x^+^x+--=b4:+-z=z—-  (Rule  I.)  : 


:.x+^ 


4~"  4 
3 


_     /  225_15 
~  V     4  ~  2  ' 


25 3 

and  x= — —^=6  miles  an  hour  for  B, 

rr+3=9  "  A. 

Prob.  6.  Some  bees  had  alighted  upon  a  tree ;  at  one 
flight  the  square  root  of  half  of  them  went  away ;  at  another 
f  ths  of  them ;  two  bees  then  remained.  How  many  alighted 
on  the  tree  ?* 

Let  2a;^=the  No.  of  bees 

then  x+  ——4-2=2^^ 

or'9a;+16a;^+18=18x2 
.-.  18ir'-16^2~9a:=:18, 
.      or  2a;'— 9^:^:18. 
(Rule  II.)  Multiply  by  8 

16^^-72^=:  144. 

*  This  question  is  taken  from  Mr.  Strachey's  translation  of  tlie 
Bija  Ganita ;  and  the  several  steps  of  the  operation  will,  upon  com- 
parison, be  found  to  accord  with  the  Hindoo  method  of  solution,  as  it 
stands  in  that  translation,  p.  62. 


96  -  ALGEBRA. 

Add  81;   then  16a;^~72rr+81=225,  i 

or  4:X—  9  ±=15  ; 
.-.  4x=15+9  =24, 

24 

and  x=—-z=6i 

.-.  V=72,  No.  of  bees. 

Prob.  7.  To  divide  the  number  83  into  two  such  parts  that 
their  product  shall  be  162.  Ans.  27  and  6. 

Prob.  8.  What  two  numbers  are  those  whose  sum  is  29, 
and  product  lOO  ?  Ans.  25  and  4. 

Prob.  9.  The  difference  of  two  numbers  is  5,  and  ;|th  part 
of  their  product  is  26.^  What  are  the  numbers  1 

A71S,  13  and  8. 

Prob.  10.  The  difference  of  two  numbers  is  6 ;  and  if  47 
.  be  added  to  twice  the  square  of  the  less,  it  will  be  equal  to  the 
square  of  the  greater.     What  are  the  numbers  ? 

Ans.  17  and  11. 

Prob.  11.  There  are  two  numbers  whose  sum  is  30;  and 
Jd  of  their  product  jpZw5  18  is  equal  to  the  square  of  the  less 
number.     What  are  the  numbers'?  Ans.  21  and. 9. 

Prob.  12.  There  are  two  numbers  whose  product  is  120. 
If  2  be  added  to  the  less,  and  3  subtracted  from  the  greater, 
the  product  of  the  sum  and  remainder  will  also  be  120. 
What  are  the  numbers'?  Atis.  15  and  8. 

Prob.  13.  A  and  B  distribute  £1200  each  among  a  certain 
number  of  persons :  A  relieves  40  persons  more  than  B,  and 
B  gives  £5  apiece  to  each  person  more  than  A.  How  many 
persons  were  relieved  by  A  and  B  respectively  1 

Ans.  120  by  A,  80  by  B. 

Prob.  14.  A  person  bought  a  certain  number  of  sheep 
for  £120.  If  there  had  been  8  more,  each  sheep  would 
have  cost  him  10  shillings  less.  How  many  sheep  were 
there  ?  Ans.  40. 

Prob.  15.  A  person  bought  a  certain  number  of  sheep 
for  £57.  Having  lost  8  of  them,  and  sold  the  remainder  at 
8  shillings  a-head  profit,  he  is  no  loser  by  the  bargain.  How 
many  sheep  did  he  buy '?  Ans.  38. 


QUADRATIC  EQUATIONS.  9T 

pROB.  16.  A  and  B  set  off  at  the  same  time  to  a  place  at 
the  distance  of  300  miles.  A  travels  at  the  rate  of  one  mile 
an  hour  faster  than  B,  and  arrives  at  his  journey's  end  10 
hours  before  him.  At  what  rate  did  each  person  travel  per 
.our  ?  Ans,  A^ravelled  6  miles  per  hour. 

Br      "        5  " 

Prob.  17.  To  divide  the  number  16  into  two  such  parts, 
that  their  product  shall  be  equal  to  70. 

Let  a;=one  part, 
then  16— a; = the  other  part 
Hence  x  {16— x)  or  16a;— a;^=70. 

Transpose,  and  x'^—lQx=—  70.  ♦ 
Complete  the  square, 

a;2_i6a;+64=-70+64=-6,  

.-.  x-S=±V^,  or  x=S+V—6* 

Prob.  18.  To  divide  the  number  20  into  Wo  such  parts, 
that  their  product  shall  be  105   -    -    -     a;=10+.\/— 5. 

Prob.  19.  To  resolve  the  number  a  into  two  such  factors, 
that  the  sum  of  their  nth.  powers  shall  be  equal  to  b. 

Let  ar=one  factor,f 

then  -==:the  other  factor. 

x 

a" 
Hence  a;"H — -=5, 

or  ic*'*4-a"=^"; 
.•.  a;**— 5i»**=— a*. 

*  It  is  very  well  known  that  the  greatest  product  which  carj  arise 
ifrom  the  multiplication  of  the  two  parts  into  which  any  given  number 
may  be  divided,  is  when  these  two  parts  are  equal ;  the  greatest  pro- 
duct therefore,  which  could  arise  from  the  division* of  the  number  16 
into  two  parts,  is  when  each  of  them  is  8  ;  hence,  in  requiring  "  to 
divide  the  number  16  into  two  such  parts"  that  their  product  should  be 
*70,"  the  solution  of  the  question  is  impossible. 

f  By  factors  are  here  meant  the  two  numbers  which  being  multi- 
plied together  shall  generate  the  given  number ;  if  therefore  x  =  on« 

factor,  -  must  be  the  other  faot3r,  for  xX-=a. 

X  ^  X 

9 


98  ALGEBRA. 

By  Rule  II. 

and  2a;"— 6  =  ^6^— 4a», 


h  +  y/¥^Aa* 


or  2a;"= 5  +  v^6^-4a\  and  a;"=  „ 


...  x=\/bJ-/_^-^ 


Prob.  20.  To  resolve  the  number  18  into  two  such  fac- 
tors, that  the  sum  of  their  cubes  shall  be  243. 

#  Ans,  6  and  3. 


ON  THE    SOLUTION   OF    QUADRATIC    EQUATIONS  CONTAINING  TWO 
UNKNOWN    QUANTITIES. 

The  solution  of  equations  with  two  unknown  quantities,  in 
which  one  or  both  these  quantities  are  found  in  a  quadratic 
form,  can  only,  in  particular  cases,*  be  effected  by  means  of 
the  preceding  Rules.  Of  these  cases,  the  two  following  are 
very  well  known. 

Case  I. 

70.  "  When  one  of  the  equations  by  which  the  values  of 
the  unknown  quantities  are  to  be  determined,  is  a  simple 
equation;"  in  which  case,  the  Rule  is,  "  to  find  a  value  of  one 
of  the  unknown  quantities  from  that  simple  equatioUj  and 
then  substitute  for  it  the  value  so  found,  in  the  other  equation ; 
the  resulting  equation  will  be  a  quadratic,  which  may  be 
solved  by  the  ordinary  Rules." 

*  The  most  complete  form  under  which  quadratic  equations  con- 
taining two  unknown  quantities  could  be  expressed,  is  this : 

a  x'^-\-h  y'^-\-c  xy-\-d  ar-j-  e  y=m 

a'x^-\-h'y'^-\-^xy-\-d'x-\-e''y^=m' \  but  the  general 
solution  of  these  equations  can  only  be  efifected  by  means  of  equations 
of  higher  dimensions  than  quadratics. 

70,  There  are  two  well-known  cases,  which  adinit  of  solution  by  the 
preceding  rules ;  state  them,  aud  the  rules  employed  for  reducing  the  two 
equations  to  one  quadratic  of  the  usual  form. 


QUADKATIC  EQUATIONS.  99 

Ex.  1. 

Let  aj+2y=7,  [  to  find  the  values  of  x  and  y. 

and  x^+^xy—y^=^o  J 

From  1st  equation,  a;=7— 2y,  /.  a;^=49— 28^+4?/* ; 

Substitute  these  values  for  x  anfx'^  in  the  2d  equation, 

then  49-.28y+4y^+.2l2/-6y^-/=23, 

or  32/^+7y==49~23=:26. 

By  Rule  II.     36/+84y+49=312+49=:361, 

/.  6y+7=19 

6?/=19-7=12,  ory=:2 

a;=7-2y=7-4=3. 

Ex.  2. 

2^+y_        ) 

"3       ~        >•  to  find  the  values  of  x  and  y. 

and  3a;y=210  ; 

From  1st  equation,  2x+y=21l ; 

.-.  2a;=27-.y. 

#  ^        27-y 

and  x= — - — 

27 —v 
Hence,  3a;y=3  X  — ^  Xy =210, 

or3x  (27-y)xy=420 
81y-3?/2=420 
27y-  2/^=140; 
or  2/^-27?/= -140. 
By  Rule  II.,  42/'-108y +729=729-560= 169  r  ^M 

.-.  2y-27=13,  or  y=?I±l?=20,       * 

.        27-20     _, 
and  a;= — - — =3^. 

Ex.3. 

There  is  a  certain  number  consisting  of  two  digits.  The 
left-hand  digit  is  equal  to  3  times  the  right-hand  digit ;  and 
if  12  be  subtracted  from  the  number  itself,  the  remainder 


/ 


100  *    ALGEBKA. 

will  be  equal  to  the  square  of  the  left-hand  digit.     What  is 
the  number  ? 

Let  X  be  the  left>hand  digit, )  then,  by  Art.  61,  10^+y 
and  y  the  other  ;  j      is  the  number. 

Hence,  ir=3v )  ,      .,  ,. 

and  10a:+y-12=  x^  \  ^^  *^^  ^^^^^^^^\ 

'st&on"  \  30y+y-12=9y^  (for  10a.=30y,  and  a:'= V)  ; 
V-31y=-12; 
,31  12 

„    ^         ^     ,     31    .  961     961     12    961-432    529 
By  Rule  I.,  2,^_2,+_=_-_=_^^^^_. 

TT  31     23  54     . 

Hence,  y~jg=-;  or  y=-=3, 

rr=3y=9; 
and  consequently  the  number  is  93. 

Ex  4.       Let  2x—Sy=  1  )  .    ^   .  ^,        i        %         . 

Ans.  x=5^  3/=3. 

Ex.  5.  There  are  two  numbers,  such,  that  if  the  less  be 
taken  from  three  times  the  greater,  the  remainder  will  be 
35 ;  and  if  four  times  the  greater  be  divided  by  three  times 
the  less  plus  one,  the  quotient  will  be  equal  to  the  less  num- 
ber.    What  are  the  numbers?  Ans,  13  and  4. 

Ex.  6.  What  number  is  that,  the  sum  of  whose  digits  is 
15,  and  if  31  be  added  to  their  product^  the  digits  will  be 
inverted?  Ans,  78. 

Case  II 

71.  When  a;',  y^,  or  xy^  is  found  in  every  term  of  the  two 
equations,  they  ajsume  the  form  of 

ax^'\-h  xy'\-cy^z=zd^ 

a'x^-\-Vxy-\-c'y'^z=zd' \  and  their  solution 
may  be  effected  :— as  in  the  following  Examples : 


QUADRATIC   EQUATIONS.  101 

Ex.  1. 

Let2a;^+3a;y+3/^=20 


^  20 

Assume  x=vi/,  then22;y +3v3/'+y'=20,  or  y^=  ^y'^-i-Sv+V 

41 

and  5i;y +4y2=41,  or  y''=^:j^^', 

which  reduced  is,  62;^— 41  v=  — 13; 


41v         13 


„    ^        ^      ,     4I1;  .  1681     1369 
ByRuLEl.,.'-- ^+-j5^=-j45:, 

41     +37  41+37     13      1 

-^-12=l2-'"'^=-^r"=2"'3- 

'  1    ^       ,        41  41       369     ^ 

Let  .=-,  thenf^^-^=^^=—=^9,  or  y=3, 

a;=i;y=ix3=l.,      , 

Ex.  2.  ,/'-^  •^^'  '    '    •        '      ' 

What  two  numbers  are  those\  who§e;  suni  ^n/ultipU'^d  b^r  >  ^ 
the  greater  is  77?  and  whose  diffefen'ce  ^tdtiplie'd  by  fcteiess  ' 
is  equal  to  12? 

Let  ir= greater  number, 
yr=less. 
Then  {x+y)Xx=x^+xi/==17, 
and  (a;— y)Xy=^y— y*=12. 
Assume  x=v2/; 

Then  vy+vy^=17,  (  ^^  ^      v^-\-v  ' 

and  V— y'=12  C        ,       12 
^      ^  3  or  y^= -. 

12        77 
Hence,  — -^-—-^ 


9* 


102'  ALaEBRA. 

OT  l2v^+l2v=lf7v-77 ', 

which  gives  v^——v  =  ——- 

■      .     65    .  4225^.529 

and  V* v-^ :^  — : 

12  ^576      576' 


65+23     88  or  42     11      7 


-or-. 


24  24  3      4 

Either  value  of  v  will  answer  the  conditions  of  the  question ; 

but  take  z;=-;  then  y^=^^--^=^_-^=^r:-^=:-=16, 

and  y=4, 

7 
a;=:vy=-  x4=7. 

Hence,  the  numbers  are  4  and  7. 

Ex.  3.  Find  two  numbers,  such,  that  the  square  of  the 
greater  minus  the  square  of  the  less  may  be  56 ;  and  the 
square  of  the  less  plus  Jd  their  product  may  be  40. 

Arts,  9  and  5. 
Ex.  4.     There  are  two  numbers,  such,  that  3  times  the 
square  of  the  greater  ^Zw5  twice  the  square  of  the  less  is  110; 
■  and  half  their  pio4uct  ^Zw5  the  square  of  the  less  is  4.  What 
afe-  tfie  nunibetst  ^  Ans,  6  and  1. 


CHAPTER  VI. 

ON   ARITHMETICAL,    GEOMETRICAL,    AND   HARMONICAL 
PROGRESSIONS. 

72.  If  a  series  of  quantities  increase  or  decrease  by  the 
continual  addition  or  5w6^rac^{o;i  of  the  same  quantity,  then 
those  quantities  are  said  to  be  in  Arithmetical  Progression. 

*  For  a  great  variety  of  questions  relating  to  quadratic  equations 
which  contain  two  unknown  quantities,  see  Bland's  Algebraical  Proh^ 
kms. 


ARITHMETICAL   PROGRE^biO^,  103 

Thus  the  numbers,  1,  2,  3,  4,  5,  6,  6oii.  (which  increase  by 
the  addition  of  1  to  each  successive  teim),  and  the  numbers 
21,  19,  17,  15,  13,  11,  &c.  (which  decrease  by  the  suhtrao 
Hon  of  2  from  each  successive  term),  are  in  arithmetical  pro- 
gression. 0 

73.  In  general,  if  a  represents  the  first  term  of  any  arith- 
metical progression,  and  d  the  cominon  difference^  then  may 
the  series  itself  be  expressed  by  a,  a-\-dj  a+2c/,  a-f  3c^,  a-^^d^ 
&c.,  which  will  evidently  be  an  increasing  or  a  decreasing  one, 
according  as  c?  is  positive  or  negative. 

In  the  foregoing  series,  the  coefficient  of  d  in  the  second 
term  is  one;  in  the  third  term  it  is  two;  in  i\iQ^  fourth  it  is  ^^ree, 
&c.,  i.  e.,  the  coefficient  of  d  in  any  term  is  always  less  by 
unity  than  the  number  which  denotes  the  place  of  that  term 
in  the  series.  Hence,  if  the  number  of  terms  in  the  series 
be  denoted  by  (7^),  the  nth.  or  last  term  in  the  progression 
will  be  a+(n— l)c/;  and,  if  the  nth  term  be  represented  bj 
I ;  then 

^=a+(n  — l)c?. 

Ex.  1.  Find  the  50th  term  of  the  series,  1,  3,  5,  7,  &c. 
Here  a=  1  )  .-.  Z=l-|-(50-l)  2 
d=  2\       =1+49x2 
n=50 )        =99. 

Ex.  2,  Find  the  12th  term  of  the  series  50,  47,  44,  &o. 
Here  a=     50  )  .-.  ^=50-f  (12-1)X -3. 
d=-  SV       =50-11x3 
n=     12)        =17. 

Ex.  3,  Find  the  25th  term  of  the  series,  5,  8,  11,  &c. 

Ans,  77. 
Ex.  4.        "         12th       «        «        «     15,  12, 9,  &c. 

Ans,  —18. 
Ex.  5.    Find  6  arithmetic  means  (or  intermediate  terms) 
between  1  and  43. 

Here  the  number  of  terms  is  8,  namely,  the  6  terms  to  be 
inserted,  and  the  2  given  terms,  and  consequently 


73.   What  is  an  arithmetiG  progression?  Give  an  example  of  a  series  of 
quantities  in  arithmetical  progression. 


104  ALGEBRA. 

a=   1  )  Buta+  {n^l)  d=l 
Z=43V  .•.1+76^=43; 

n=  8  )  .-.  d=6. 

And  the  means  required  are  7,  13,  19,  25,  31,  37. 

Ex.  6.    Find  7  arithmetic  means  between  3  and  59. 

Ans.  10,  17,  24,  31,  38,  45,  52. 

Ex.  7.    Find  8  arithmetic  means  between  4  and  67. 
Ex.  8.    Insert  9  arithmetic  means  between  9  and  109. 

74.  *  Let  a  be  the  ^rst  term  of  a  series  of  quantities  in 
arithmetic  progression,  d  the  common  difference,  n  the  num- 
ber of  terms,  I  the  last  term,  and  S  the  sum  of  the  series : 
Then 

S=^a-\'{a+d)-\-{a+2d)-^  -  -  -  +^ 
and,  writing  this  series  in  a  reverse  order, 

S=l+{l-d)+{l-2d)+    --.+«. 
These  two  equations  being  added  together,  there  results 
2  S={a+l)  +  {a+l)  +  {a+l)+  -  -  -  +(a+0 
=  (a+Z)X^,  since  there  are  n  terms; 
...  S={a+l)l  ....  (1). 

Hence  it  appears  that  the  sum  of  the  series  is  equal  to  the 
sum  of  the  first  and  last  terms  multiplied  by  half  the  number 
of  terms : 

And  since  l=a+{n—l)  d; 

.\S=ha+{n^l)dl^ (2). 

From  this  equation,  any  three  of  the  four  quantities  a,  rf, 
n,  8,  being  given,  the  fourth  can  be  found. 

Ex.  1.    Find  the  sum  of  the  series  1,  3,  5,  7,  9,  11,  &c. 

continued  to  120  terms. 


Herea=     1 Y...  ^=  |  2a+(n-l)  c?[  x| 

=  i2xl+(120~l)2J.  X 


d=     2  r  ,  ^      120 

7i=120j         -  ]  •'*''"     *^'^f'^2 

(2+ 1 19  X  2)  X  60=240X60= 14400. 


ARITHMETICAL  PROGRESSION. 


105 


Ex.  2.    Find  the  sum  of  the  series  15,  11,  7,  3,-~l,  — 5, 

&;c.,  to  20  terms. 


=  |2xl5  +  (20~l)x-4  Ix 


20 
2 


=  (30-19x4)xl0 
=  (30-76)  X 10 
=  —46X10= -460. 

Ex.  3.  Find  the  sum  of  25  terms  of  the  series  2,  5,  8,  11, 
14,  &c.  Ans.  950. 

Ex.  4.  Find  the  sum  of  36  terms  of  the  series  40,  38,  36, 
34,  &c.  Ans,  180. 

Ex.  5.  Find  the  sum  of  150  terms  oif  the  series  ^,  |^,  1,  j, 
f ,  2,  I,  &c. 

Herea=i     1   .'.  >^=  |  2a+(.-l)  c^[  ^ 

^=1       I         =|2XK(150-I)xij^ 

n=150j  =  g+l|?)75=l|lx75=3775. " 

Ex.  6.  Find  the  sum  of  32  terms  of  the  series  1,  1^,  2, 
2^,  3,  &c.  Ans.  280. 

PROBLEMS. 

pROB.  1.  The  sum  of  an  arithmetic  series  is  1240,  common 
difference  —  4,  and  number  of  terms  20.  What  is  the  Jirst 
term  ? 

Here  5=  1240 1  .-.    S=  |  2a+ (ti-I)  cZl  ^ 

^=-"  '*hl240=i2a+(20-l)x  ' '^\^ 

w=     20  J  =  (2a-19x4)10 

124=2a-76; 
.-.  2a= 124+76=200, 
and.-.  a=100. 
Hence  the  series  is  100,  96,  92,  &o. 


106  ALGEBRA. 

Prob.  2.  The  sum  of  an  arithmetic  series  is  1455,  \hQ  first 
term  5, -and  the  number  of  terms  30.  What  is  the  common 
difference  ? 

Here  .S^  1455 ' 


a=       5 
nz=z     30  J 


|<Ja+(n-iyj.^=>Sf       - 
'.-.  i2x5+(30~l)cZl^=1455 


a=     7 
d=     2) 


(10+29^)15=1455; 
Dividing  both  sides  by  15,  10+29c?=97, 

29cZ=87; 
.-.  c?=3. 
Hence  the  series  is  5,  8,  11,  14,  &c. 

Prob.  3.  The  sum  of  an  arithmetic  series  is  567,  ihe  first 
term  7,  the  common  difference  2.    Find  the  number  of  terms. 

Here  5=567l  .-.  since  i  2a+  (n-1)  d  1 1=5 

■"  '^  i2x7+(7i-l)2l|=567 

7i^+6?i=567. 
Completing  the  square,  ^'+6^+9=576, 

.    Extracting  the  square  root,   7i+3=_+24; 

.-.  w=21  or  —27. 

Prob.  4.  The  sum  of  an  arithmetic  series  is  950,  the  com- 
mon difference  3,  and  number  of  terms  2b.  What  is  theirs/ 
term?  Ans,  2. 

Prob.  5.  The  sum  pf  an  arithmetic  series  is  165,  the^rs^ 
term  3,  and  the  number  of  terms  10.  What  is  the  common 
difference?  Ans.  3. 

Prob.  6.  The  sum  of  an  arithmetic  series  is  440,  first 
term  3,  and  common  difference  2.  What  is  the  number  of 
terms?  Ans.  20. 

Prob.  7.  The  sum  of  an  arithmetic  series  is  54,  \hQ  first 
term  14,  and  common  difference  —2.  What  is  the  number  of 
terms  ?  Ans.  9  or  6. 


ARITHMETICAL  PROGRESSION.  107 

Prob.  8.  A  traveller,  bound  to  a  place  at  the  distance  of 
198  miles,  goes  30  miles  ihejirst  day,  28  the  second,  26  the 
third,  and  so  on.  In  how  many  days  will  he  arrive  at  his 
journey's  end  ? 

Here  is  given  a=       SO  )      ^ 

d=:   — •  2  >-  to  find  the  number  of  terms. 
S=     198     . 


Now  i.2a+  {n-l)dl'^=S, 

i2x30+(7i-l)x~2l^=198, 
(31-^)^=198, 


or,  71^—3171=:  — 198, 

31       ,  13 

"-2=±2-' 

31  ,  13     ^^      ^ 
/.  7i=— +— =22or9. 

To  explain  the  apparent  difficulty  arising  from  the  two 
positive  values  of  n,  which  gives  us  two  different  periods  of 
the  traveller's  arrival  at  his  journey's  end,  we  must  observe, 
that  if  the  proposed  series  30,  28,  26,  &c.,  be  carried  to  22 
terms,  the  16th  term  will  be  nothing,  and  the  remaining  six 
terms  ^ill  be  negative  ;  by  which  is  indicated  the  rest  of  the 
traveller  on  the  16th  day,  and  his  return  in  the  opposite 
direction  during  the  six  days  following ;  and  this  will  bring^^ 
him  again,  at  the  end  of  the  22d  day,  to  the  same  point  at 
which  he  was  at  the  end  of  the  9th,  viz.,  198  miles  from  the- 
place  whence  he  set  out. 

Prob.  9.  How  much  ground  does  a  person  pass  ov^  ih' 
gathering  up  200  stones  placed  in  a  straight  line,  at  intervals 
of  2  feet  from  each  other ;  supposing  that  he  brings  each  stone^ 
singly  to  a  basket  standing  at  the  distance  of  20  yards  from, 
the  first  stone,  and  that  he  starts  from  the  spot  where  the- 
basket  stands  ? 

It  is  evident  that  the  space  passed  over  by  this  person;  will^ 
be  twice  the  sum  of  an  arithmetic  series,  whose  j'^rs^  term  is 


108  ALGEBRA. 

20  yards  (i.  e.  QfOfeet)^  common  difference  2  feet,  and  number 
of  terms  200.  • 

Herea=60),^^|2,+  („_l),J^« 

^   ^=200)         =(12i^+398)Xl00. 

==518x100=51800  feet. 

feet,  miles,         furlongs,  feet. 

Hence  the  distance  required^  103,600=    19     -    4    -    640. 

Prob.  10.  a  person  bought  47  sheep,  and  gave  1  shilling 
for  the^r^^  sheep,  3  for  the  second^  5  for  the' third,  and  so  on. 
What  did  all  the  sheep  cost  him?  Ans,  £110  9^. 

Prob.  11.  A  gentleman  began  the  year  by  giving  away  a 
farthing  \he  first  day,  a  halfpenny  the  second,  three  farthings 
the  third,  and  so  on.  What  money  had  he  disposed  of  in 
charity  at  the  end  of  the  year?  Ans,  £69  11 5.  6fc?. 

Prob.  12.  A  travels  uniformly  at  the  rate  of  6  miles  an 

hour,  and  sets  off  upon  his  journey  3  hours  and  20  minutes 

before  B ;  B  follows  him  at  the  rate  of  5  miles  the  first  hour, 

,  6  the  second,  7  the  third,  and  so  on.    In  how  many  hours  will 

B  overtake  A?  Ans,  In  8  hours. 

Prob.  13.  There  is  a  certain  number  of  quantities  in 
arithmetic  progression,  whose  common  difference  is  2,  and 
whose  sum  is  equal  to  eight  times  their  number ;  moreover, 
If  13  be  added  to  the  second  term,  and  this  sum  ba>  divided 
by  the  number  of  terms,  the  quotient  will  be  equal  to  \h.e  first 
term.    What  are  the  numbers  % 


Jn  the  expression  2a-\-(ii—\)(i)<,-,  substitute  x  for  a,  2  for  5, 

■and  y  for^i,  and  it  becomes  2a:+(y  — l)2X^(=^y+2/^^y), 
ibr  the  sum  of  the  series. 

By  the  problem,  xy+y^—y=Sy,  or  y=9— a?, 

^  a;4-2+13 
and =ix. 


GEOMETEIC  PROaRESSION.  109 

Hence,  — ■ — =x,  or  ic^— 8^=  — 15: 

9 — X 

.\  x'—Sx+16=l6-16  =  l, 

andic— -4=  +  l ;  .*.  x=5  or  3, 
^      y=9— ir        —4  or  6. 
From  which  it  appears  that  there  are  two  sets  of  numbers 
which  will  answer  the  conditions  required;  viz.,  5,  7,  9,  11,  or 
3,  5,  7,  9,  11,  13. 

Prob.  14.  There  is  a  certain  number  of  quantities  in 
arithmetic  progression,  whose  ^rs^f  term  is  2,  and  whose  sum 
is  equal  to  8  times  their  number ;  if  7  be  added  to  the  third 
term,  and  that  sum  be  divided  by  the  number  of  terms,  the 
quotient  will  be  equal  to  the  common  difference.  What  are 
the  numbers]  Ans.  2,  5,  8,  11,  14. 

ON    GEOMETRIC    PROGRESSION. 

75.  If  a  series  of  quantities  increase  or  decrease  by  the 
continual  multiplication  or  division  by  the  same  quantity,  then 
those  quantities  are  said  to  be  in  Geometrical  Fvogression. 
Thus  the  numbers,  1,  2,  4,  8,  16,  &;c.  (which  increase  by  the 
continual  multiplication  by  2),  and  the  numbers  1,  i  i  ^, 
&c.  (which  decrease  by  the  continued  division  loj  3,  or 
multiplication  by  ^),  are  in  Geometrical  Progression. 

76.  In  general,  if  a  represents  the  Jirst  term  of  such  a 
series,  ^nd  r  the  common  multiple  or  ratio,  then  may  the 
series  itself  be  represented  by  a,  ar,  ar"^,  ar^,  ar\  &;c.,  which 
will  evidently  be  an  increasing  or  decreasing  series,  according 
as  r  is  a  whole  number  or  2i.  proper  fraction.  In  the  foregoing 
series,  the  index  of  r  in  any  term  is  less  by  unity  than  the 
number  which  denotes  the  place  of  that  term  m  the  series. 
Hence,  if  the  number  of  terms  in  the  series  be  denoted  by 
(n),  the  last  term  will  be  ar"^"^, 

77.  From  the  series  given  in  the  two  preceding  articles  it 
is  evident,  by  mere  inspection,  that  the  common  ratio  can  be 
found  by  dividing  the  second  term  by  Xhe  first,  or  by  dividing 
any  term  by  that  which  precedes  it. 

75.  Define  a  geometrical  progression,  and  give  an  example. — 77.  How  is 
the  common  ratio  of  a  series  of  numbers  in  geometrical  progression  found  ? 


110  ALGEBRA. 

Ex.  1.  Find  the  common  ratio  of  the  geometrical  progression 
1,  2,  4,  8,  &c. 

Here  the  common  ratio  =~=2. 

^  2  4    8 

Ex.  2.  Find  the  common  ratio  of  the  series  -,  -,  — ,  &;c. 

4  2     2 

In  this  example  the  common  ratio  =  -H--=^. 

y    o    o 

5  3    9 

Ex.  3.  Find  the  common  ratio  in  the  series  -,  1,  -,  -— ,  &c. 

o       5  25 

Ans.  -. 
5 

78.  Let  S  be  the  sum  of  the  series  a,  ar^,  ar^,  (fee,  then 
a-{-ar+ar'^+ar^+k>Q,.  -  -  -  ar""'"^ + ar'^'^ =.  S, 

Multiply  the  equation  by  r,  and  it  becomes 

Subtract  the  upper  equation  from  the  lower^  and  we  have, 
ar''—a=zrS^S^ox{r—\)  S=ar''—a', 

ar^     cCr 

and  therefore,  S= — . 

r— 1 

If  r  is  a  proper  fraction,  then  r  and  its  powers  are  less 

than  1. 

For  the  convenience  of  calculation,  therefore,  it  is  better 

a '~~  ar^ 
in  this  case  to  transpose  the  equation  into  S  =  — ,  by" 

multiplying  the  numerator  and  denominator  of  the  fraction 

ar^'^a  , 

T-  by  —1. 

79.  If  I  be  the  last  term  of  a  series  of  this  kind,  then 

l=ar*^^,  ,\rlz=:ar'^\  hence  0=1 7-)= r-.     xrom   this 

\r— 1/      r—l 

equation,  therefore,  if  any  three  of  the  four  quantities  >S',  a,  r, 

Z,  be  given,  the  fourth  may  be  found. 


78.  What  is  the  expression  for  the  sum  of  n  terms  of  a  series  of  mimbera 
in  geometrical  progression? 


GEOMETRIC   PROGRESSION.  Ill 

Ex.  1. 

Find  the  sum  of  the  series  1,  3,  9,  27,  &c.  to  12  terms. 
ar--a     lx3^'-l 


,s=:- 


_8P-1 
""      2     • 
531441-1     531440 


=265720. 


~         2  2 

Ex.  2. 

2    4     8' 
Find  the  sum  often  terms  of  the  series  l+o+5+^,  <^c. 

o    y    i4  / 


a=  1 

2 

'=  3 

n=10 


1-r  ~  ^_2  3-2  W 


3 

,,      ,^V'    2''      1024 

N0W(-; 


■•■'-©'■= 


3'^~59049' 

1024  _58025 
"59049"  59049' 
3 X 58025 _ 174075 
^"""^  '^"~59049~~"59049"- 


Ex.  3.    Find  the  sum  of  7  terms  of  the  series,  1,  3,  9,  27, 
81,  &c.  Ans.  1093. 

Ex.  4.  Find  the  sum  of  1,  2,  4,  8,  16,  &c.  to  14  terms. 

Ans.  1638eS. 

Ex.  5.  Find  the  sum  of  1,  -,  -,  — ,  &;c.  to  8  terms. 

O      9     y&7 

,      3280 

^"^•2187- 
Ex.  6.  Find  three  geometric  means  between  2  and  32. 
Here  a=  2  )  And  ar"-^=Z 
Z=32  ^        .-.  2r^  =32, 
71=  5)  r*  =16, 

.-.  r   =  2. 
And  the  means  required  are  4,  8,  16. 


112  ALGEBBA. 

Ex.  7.  Find  two  geometric  means  between  4  and  256. 

Ans.  16  and  64. 

Ex.  8.  Find  three  geometric  means  between  ^  and  9. 

^.  Ans.  ^,  1,  3. 

Ex,  9.  Find  a  geometric  mean  between  a  and  I. 
Let  X  be  the  geometric  mean  required ; 
Then  a,  x,  Z,  are  three  terms  in  geometric  progression, 

-  X       I 

and  -=  - 
a     X 

or  x^=:al 
.*.  xziz-y/aL 

EiT  10.  What  is  the  geometric  mean  between  16  and  64? 

Ans.  32. 
Ex.  11.  Insert  four  geometric  means  between  ^  and  81. 

Ans.  1,  3,  9,  27. 

PROBLEMS. 

Prob.  1.  Find  three  numbers  in  geometric  progression; 
such  that  their  s^im  shall  be  equal  to  7  ;  and  the  sum  of  their 
squares  to  21. 

Let  -,  0?,  xy^  be  the  numbers.  Then  by  the  problem, 
-+x+xy     =7  -  -  (1) 
^--^+x''+xY=2\  .  -  (2) 

From  equation  (1),     xi — l-l+yj  =  7 

/I      2  \ 

•.  by  squaring,  a;'/— +-+3+2y+2/M=49 

From  (2)  x'O-^       +1         +y')==21 

.-.  by  subtraction,    x4--+2+2y\       =28, 

or  14a: =28 

.-.  X=:    2. 


GEOMETKIC  PROGRESSION.  113 


This  value  ofx  being  inserted  in  (1), 
1  7 


•••^-22'+l2J  =16-1=16 


5±3    „      1 

Hence,  the  numbers  are  1,  2,  4;  or  4,  2,  1. 

Prob.  2.  There  are  three  numbers  in  geometric  progression 
whose  product  is  64,  and  sum  14.    What  are  the  numbers  ? 

Ans.  2,  4,  8 ;  or  8,  4,  2. 

Prob.  3.  There  are  three  numbers  in  geometric  progression 
whose  sum  is  21,  and  the  sum  of  their  squares  189.  What  are 
the  numbers?  Ans.  3,  6,  12. 

Prob.  4.  There  are  three  numbers  in  geometric  progression; 
the  sum  of  the  Jirst  and  last  is  52,  and  the  square  of  the  mean 
is  100.  What  are  the  numbers?  Ans.  2,  10,  50. 

Prob.  5.  There  are  three  numbers  in  geometric  progression, 
whose  sum  is  31,  and  the  sum  of  thej^rs^and  last  is  26.  What 
are  the  numbers?  Ans.  1,  5,  25. 


ON  THE  SUMMATION  OF  AN  INFINITE  SERIES  OF  FRACTIONS  IN 
GEOMETRIC  PROGRESSION ;  AND  ON  THE  METHOD  OF  FINDING 
THE  VALUE  OF  CIRCULATING  DECIMALS. 

79.    The  general  expression  for  the  sum  of  a  geometric 

series  whose  common  ratio  (r)  is  a  fraction,    is  (Art.  78) 

a  ~~  ar^ 
S  z=— .    Suppose  now  n  to  be  indefinitely  gi-eat,  then 

r*  (r  being  a  proper  fraction)  will  be  indefinitely  small,'^  so 

*  When  r  is  a  proper  fraction,  it  is  evident  that  r"  decreases  as  n 

increases;   let  r=:i  for  instance,  then  r«=ji^,  r^=^^,  ri=^^M., 

and  when  n  is  indefinitely  great,  the  denominator  of  the  fraction 
becomes  so  large  with  respect  to  the  numerator,  that  the  value  of  the 
fraction  itself  becomes  less  than  any  assignable  quantity. 

10* 


114  ALGEBRA. 

that  ar""  may  be  considered  as  nothing  with  respect  to  a  in 
the  numerator  a—ar""  of  the  fraction  expressing  the  value 
of /S^;  the  limit^  therefore,  to  which  this  value  of /S  approaches, 

when  the  number  of  terms  i^infinite,  is . 

1 — r 

Ex.  1. 

Find  the  sum  of  the  series  1 +0+7+0?  ^^'  ^^  infinitum, 
Herea  =  l)       c^       ^  1  ^        ^ 

Ex.  2. 

Find  the  value  of  k+ 97 +79^  +  ^0.  ad  infinitum. 


Here  a=- 
5 


5  '  25  '  125 

1 

5  1        1 


,=i 


,'.S=: 


1     5-1     4 
5j  5 

Ex.  3.  Find  the  value  of  1  +— +— +^+&c.  ad^  infinitum, 

Ans,  —. 

3  9     27 

Ex.  4.  Find  the  valueof  l+--r-+ 777+^-7 +&c.  ad  infinitum. 

4  Id     d4 

An$^  4. 
•  2       4       8 

Ex.  5.  Find  the  value  of -=-+^+t^+&;c.  ad  infinitum. 

80.  These  operations  furnish  us  with  an  expeditious  me- 
thod of  finding  the  value  of  circulating  decimals^  the  num- 
bers  composing   which   are  geometric  progressions,   w^hose 


80.  What  is  the  expression  for  the  sum  of  a  geometric  series,  when  the 
number  of  terms  is  iunnite  ? 


GEOMETRIC    PROGRESSION. 


115 


1  1  1  0 

common  ratios  are  — ,  — — ,  ,  &c.,  according  to  the  num- 

ber of  factors  contained  in  the  repeating  decimal. 

Ex.  if 

Find  the  value  of  the  circulating  decimal  .33333,  &;c.  This 
decimal  is  represented  by  the  geometric  series 
3        3  3  .3 

Tfi"^Tno"^inna"^^^''  ^^^^®  f''^^^  ^^^^  ^®  Ta'  ^^^  common 

.     1 
ratio 


10* 
Hence  «=j^, 


10' 


S=z 


10 


l-r' 


1- 


10-1 


10 


Ex.  2.  Find  the  value  of  .32323232,  &;c.  ad  infinitum. 


Here  a  = 


32  "^ 
100' 

1 
100' 


•.  s=- =■ 


32 
100 


32 


1- 


2^ 

100 


32 

'l00-l~99* 


Ex.  3.  Find  the  value  of  .713333,  &c.  ad  infinitum. 

The   series   of  fractions   representing   the   value    of  this 

71  3  3 

decimal  are  TTr^+  (geometric  series)  7777^+77^7:7777+   &c. 


Here  a=: 


100 


1000 
1 


1000  '  10000 


1000 


1- 


2 

"10      J  ^     10 

Hence  the  value  of  the  decimal  =  ( 

_107 
""150' 


1000-100     900     300 


100 


+>S 


A2L.  J_. 

7100  "^300"^ 


214 

300 


116  •      ALGEBRA. 


Ex.  4.  Find  the  value  of  .81343434,  &c.  ad  infinitum. 

34 
a  10000  34  34 


Here  a=i 

^     10000 


.     Cf, 

J         ("    ~T^~,       1       10000- 100-9900 


'■-100     J  ^     100 

K   A      ^        f,u    a'-      ^       «1    ,  c     81    ,    34       8053 
And  value  of  the  decimal  =—+S=~+^^^=^^. 

^x.  5.  Find  the  value  of  .77777,  &c.  ad  infinitum. 

,       7 
Ans.-. 

Ex.  6.  "        «  .232323  &o.  ad  infijiitum. 

A      23 

Ex.  7.  «        «  .83333,  &c.  ad  infinitum. 

Ans.^. 

Ex.  8.  «        "  .7141414,  &c.  ad  infinitum.    • 

.      707 
^'^^•990- 
Ex.  9.  "        "  .956666,  &c.  ad  infinitum. 

A      287 

^'^^•soo- 

The  value  of  a  circulating  decimal  -may  also  be  found  as 
follows : — In  Ex.  4  above, 

Let  S=     .813434  .... 
.-.  10000  5=8134.3434  .... 
and  100  S==     81.3434  .... 
•       .-.9900/^=8053 

^    8053       ,    ^ 
•'•^=9900'"^^"^"^"- 

pROB.  1.  A  body  in  motion  moves  over  1  mile  theirs/ 
second,  but  being  acted  upon  by  some  retarding  cause,  it 
only  moves  over  \  a  mile  the  second  second,  \  the  third^ 


HARMONIC   PROGRESSION.  117 

and  so  on.  Show  that,  according  to  this  law  of  motion,  the 
body,  though  it  move  on  to  all  eternity^  will  never  pass  over 
a  space  greater  than  2  miles. 

ON  HARMONIC  PbI^GRESSION. 

81.  A  series  of  quantities,  whose  reciprocals  are  in  arith- 
metic progression,  are  said  to  be  in  Harmonic  Progression. 
Thus  the  numbers  2,  3,  6,  are  in  harmonic  progression^  since 
their  reciprocals  |-,  -J,  |-,  are  in  arithmetic  progression  (—-J 
being  the  common  difference), 

2 

Ex.  1.  Find  a  harmonic  mean  between  1  and  — . 

Let  a?  be  the  mean  required : 

1      3 

Then  1,  — ,  — ,  are  in  arithmetic  progression, 

And 


1 

-1= 

3 

1 

X 

"2"" 

X 

2 

3 

• 

=  1  + 

X 

2 

5 

'2 
4 

\  X- 

'~h' 

Ex,  2.  Find  a  third  number  to  be  in  harmonic  progression 
with  6  and  4. 

Let  *  be  the  number  required : 

111  .^      . 

Then  -— ,  — -,  — ,  are  in  arithmetic  progression. 
6     4     a; 

A  :i  1    1    1    1 

•  '   a;  ~  2       6 
~6      "6" 

.•.ar=3 


118 


ALGEBKA. 


Ex.  3.  Insert  three  harmonic  means  between  9  and  3. 

The  reciprocals  of  9  and  3  are  — -  and  — ,  which  are  Xhe  first 

9  o 

and  last  term  of  an  aritkrr^ic  progression,  between  which  3 

arithmetic  means  are  to  be  inserted.    We  have  therefore, 

according  to  Art.  73 — 


And  a+(n-'l)d=l, 


1 

a=z 

— 

9 

1 

1= 

¥ 

X=z 

5 

• 

i+(5-l)fcl 


^  3      9 

"~  9      9 
""9 

Hence,  — ,  — ,  ~,  are  the  arithmetic  means  to  be  inserted 

between  —  and  •— ,  and  therefore  their  reciprocals  6,  — ,   — , 

are  the  three  harmonic  means  required. 

Ex.  4.  Find  a  harmonic  mean, between  12  and  6. 

A71S,  8. 

Ex.  5.  The  numbers  4  and  6  are  two  terms  of  a  harmonic 
progression;  find  a  third  term.  Ans.  12. 

Ex.  6.  Find  two  harmonic  means  between  84  and  56. 

Ans,  72  and  63. 
Ex.  7.  Insert  three  harmonic  means  between  15  and  3. 

Ans.~,5,-. 

82.    Let  c,   b,  c,  d,  e,  &c.,  be  a  series  of  quantities  in 

1     1     1     1     1    p 

harmonic  progression ;  then  —,  —,  — ,  --7,  — ,  &c.,  are  m  ant/i- 

a  -  0     c     a     e 


HARMONIC   PROGRESSION.  119 

rnetic   progression,    and   according   to   the   definition   of  an 
arithmetic  progression  (Art.  72),  we  have 

'    '    '    '       (1) 

(2) 
(3) 


From  (1) 


b 

a 

c 

b 

1 

1 

1 

f 

c 

b' 

"  d 

c 

1 

d 

1 

c 

_  1 

~  e 

1 
d 

(fee. 

= 

&c. 

a- 

-b 

b- 

-c 

ab 

c 

a- 

-b 

b- 

-c 

a 

c 

.', 

a 
c 

a- 

-b 

-c' 

or,  converting  this  equation  into  a  proportion-, 
a'.cwa — b'.b—c 
Similarly  from  (2)     b:d::b—c:c^d 
"         "        (3)     c:e::c—d:d'-e 
and  so  on  for  any  number  of  quantities. 

These  proportions  are  frequently  assumed  as  the  deJinitio7\ 
to  quantities  in  harmonic  progression,  and  may  be  thus 
expressed  in  words: — if  any  three  quantities  in  harmonic  pro- 
gression be  taken,  .the  first  is  to  the  third  as  the  difference 
between  the  first  and  second  is  to  the  difference  between  the  second 
and  third. 

Prob.  1.  Given  a^=b^  =z(f ,  where  a,  5,  c,  are  in  geometric 
progression.    Prove  that  ar,  yf  z,  are  in  harmonic  progression. 

y 
a^^by\  ,\a=b^   -  -  -  (1). 

y 
<f=h'^'^  ,\c=b^ (2). 

By  multiplying  (1)  by  (2)  acz=zb^     « 


120  ALGEBRA. 

But  by  geometric  progression  ac^zh^ 


y      X       z 

y     ^~2     y' 


CHAPTEE   VII. 

ON  PERMUTATIONS  AND  COMBINATIONS. 

83.  By  Permutations  are  meant  the  number  of  changes 
which  any  quantities,  a^  b,  c,  d,  e,  &c.,  can  undergo  with  re 
spect  to  their  order,  when  taken  two  and  two.  together,  three 
and  three^  (fee,  &c.  Thus  a6,  ac,  ad^  ba,  be,  bd,  ca,  cb,  cd,  da, 
db,  dc,  are  the  different  permutations  of  the  four  quantities  a, 
b,  c,  d,  when  taken  two  and  two  together;  abc,  acb,  bac,  bca, 
cab,  cba,  of  the  three  quantities  a,  b,  c,  when  t'aken  three  and 
three  together,  &;c.,  &c. 

84.  Let  there  be  n  quantities,  a,  b,  c,  d,  e,  &c. :  then,  by 
Art.  83,  it  appears  that  there  will  be  (^—1)  permutations  in 
which  a  stands  first ;  for  the  same  reason  there  will  be  (?^  — 1) 
permutations  in  which  b  stands  first ;  and  so  of  c,  r/,  e,  &c. 
Hence  there  will  be  7i  times  [n — 1)  permutations  of  the  form 
ab,  ac,  ad,  ae,  &c. ;  ba,  be,  bd,  be^  &c. ;  ca,  cb,  cd,  ce,  &c. ;  i.  e. 
"/A^  number  of  permutations  ofn  things  taken  two  and  two  is 
n(n-l)." 

85.  If  these  n  quantities  be  taken  three  and  three  together, 
then  there  will  be  n  (n  —  \)  (?i— 2)  permutations.  For  if 
(n  —  \)  be  substituted  for  n  in  the  last  article,  then  the  number 
of  permutations  of  n—\  things  taken  two  and  two  together 
will  be  (^  —  1)  ('i— 2) ;  hence  the  number  of  permutations  of 
5,  c,  d^  e^  &c.,  taken  two  and  two  together,  are  (n  —  \)  (?i— 2), 


PERMUTATIONS.  121 

and  consequently  there  are  (^  —  1)  (n— 2)  permutations  of  the 
quantities  a,  6,  c,  c/,  e,  &c.^  taken  ^Ar^g  ant/  tliree  together,  in 
which  a  may  stand  first ;  for  the  same  reason  there  are  {ji  —  V) 
(n— 2)  permutations  in  which  h  may  stand  first;  and  so  of  c, 
«?,  e^  &;c.  The  numbers  of  the  j^rmutations  of  this  kind  will 
therefore  amount  to  ti  (/i  — 1)  (n— 2). 

86.  To  find  the  number  of  'permutations  of  n  things  taken 
r  together. 

By  Arts.  85  and  86— 
The  No.  taken  two  together  z=zn{7i—\) 

"  "      three      "        =n\n—\){n—2) 

Similarly    "     four       "        =n  \n-\)  {n-2)  (/i-3) 

If  the  law,  which  is  observed  in  these  particular  cases,  be 
supposed  to  hold  generally,  that  is,  if  the  number  of  permu- 
tations of /I  things  «,  6,  c,  d^  &;c.,  taken  r  — 1  together,  be 

n  (/i  — 1)  (^—2) (7i-r+2) 

Then,  by  omitting  «,  it  is  equally  true  that  the  number  of 
permutations  of  n  — 1  things  6,  c,  d^  &;c.,  taken  r—\  together, 
will  be  by  putting  n  —  \  iov  n  in  this  last  expression 

{n  —  \)  {n—2) (?i— r+1) 

Now,  if  a  be  placed  before  each  of  these  permutations,  there 
will  be 

(n-1)  {n-2)  .....  {n-r+1) 

permutations  of  things  taken  r  together,  in  which  a  stands  first. 
It  is  clear  that  there  will  in  like  manner  be  the  same  number 
of  permutations  of  things  taken  r  together,  in  which  each  of 
the  other  things  6,  c,  d,  (fee,  stand  respective Ig  first;  and  as 
there  are  n  things,  the  entire  number  of  permutations  of  ?» 
things  taken  r  together,  will  be  the  sum  of  the  permutations 
taken  r  together  in  w^hich  the  n  things  a,  6,  c,  d,  &c.,  respect- 
ively  stand   first,    that   is,   n  times   (w—1)  [n—2) 

(n-r+1), 

or  n  {n  —  1)  [n—2) [n—r+l) 

It  has  thus  been  proved,  that  if  the  law  by  w^hich  the  ex- 
pression for  the  number  of  permutations  of  n  things  taken 
r— 1  together  is  found,  be  true,  it  is  also  true  for  the  next 
superior  number,  or  when  n  things  are  taken  r  together; 
but  the  law  of   the  expression  has  been  found  to  hold  for 

11 


122  ALGEBRA. 

the  number  of  permutations  of  n  things  taken  two  together, 
and  for  the  number  of  permutations  of  n  things  taken  three 
together ;  it  is  therefore  true  by  the  theorem  just  demonstrated 
when  the  n  things  are  taken  four  together,  and  if  true  when 
taken  four  together,  it  is  tr^  also  when  taken  j^yg  together, 
and  so  on  for  any  number  not  greater  than  n^  which  may  be 
taken  together. 

This  proof  affords  an  excellent  example  of  demonstrative 
induction^  a  method  of  reasoning  of  great  iniportance  in  the 
mathematical  sciences. 

87.  If  rr=7i,  i.  e.  if  the  permutations  respect  all  the  quan- 
tities at  once,  then  (since  n — r=^ri)  the  number  of  them  will 
be  n  (n  —  \)  (?i— 2)  &c.  ...  2.  1.  Thus,  the  number  of  per- 
mutations which  might  be  formed  from  the  letters  composing 
the  word  ''virtue'''  are  6x5x4x3x2x1=720. 

88.  But  if  the  same  letter  should  occur  any  number  of  times, 
then  it  is  evident  that  we  must  divide  the  whole  number  of  per- 
mutations by  the  number  of  permutations  which  would  have 
arisen  if  different  letters  had  occurred  instead  of  the  repetition 
of  the  same  letter.  Thus  if  the  same  letter  should  occur  twice^ 
then  we  must  divide  by  2x1 ;  if  it  should  occur  thrice^  we 
must  divide  by  3x2x  1 ;  if  p  times  by  1.2.3.  .  .  ^;  and  so  for 
any  other  letter  which  may  occur  more  than  once.  Hence 
the  general  expression  for  the  number  of  permutations  of  n 
things,  of  which  there  are  jp  of  one  kind,  r  of  another^  q  of 

.     n  (n  —  l)  in— 2)  (w— 3)  .  .  2.1       - 

another,  &c.,  &c.,  is      -. \  ^  .  ^  o   r^r^ •     J-^^s 

'        '        '  1.2.3.  .j(?X  1.2.3.. rX  1.2.3.. g 

the  permutations  which  may  be  formed  by  the  letters  com- 
posing the  word  ''easiness'''^  (since  s  occurs  thrice^  e  twice)  are 
^.7.6.5.4.3.2.1^33 
1.2.3.x  1.2 

Ex.  1.  What  is  the  number  of  different  arrangements  which 
can  be  made  of  6  persons  at  a  dinner-table? 

The  number=lx2x3x4x5x6=720. 

Ex., 2.  Required  the  number  of  changes  which  can  be  rung 
upon  8  bells. 

The  No.  of  Changes=lx2x3x4x5x6x7x8=40320. 


COMBINATIONS.  123 

Ex.  3.  With  5  flags  of  difterent  colours,  how  many  signals 
can  be  made  ? 

The  number  of  signals,  when  the  flags  are  taken — 
Singly^  are  ---^---nr  5 
Two  together  =5.4-  -  -  =20 
Three  -  -  -  =5.4.3  -  -  =  ^60 
Four  -  .  -  =5.4.3.2  -  =120 
Five     -    -    -     =5.4.3.2.1      =120 


.'.  the  total  number  of  signals  =  325 

Exl^.  How  many  permutations  can  be  formed  out  of  10  ' 
letters,  taken  5  at  a  time  1  Ans.  30240. 

Ex.  5.  How  many  permutations  can  be  formed  out  of  the 
words  Algebra  and  Missippi  respectively,  all  the  letters  being 
taken  at  once?  Ans,  2520  and  1680. 

ON  COMBINATIONS. 

89.  By  Combinations  are  meant  the  number  of  collections 
which  can  be  formed  out  of  the  quantities,  or,  6,  c,  c?,  ^,  &c,, 
taken  two  and  two  together,  three  and  three  together,  &;c.,  (fee, 
without  having  regard  to  the  order  in  which  the  quantities 
are  arranged  in  each  collection.  Thus  ab,  ac,  ad,  be,  bd,  cd, 
are  the  combinations  which  can  be  formed  out  of  the  four 
quantities  a,  b,  c,  d,  taken  two  and  two  together;  abc,  abd, 
acd,  bed,  the  combinations  which  may  be  formed  out  of  the 
same  quantities,  when  taken  three  and  three  together;  &c.,  &c. 

90.  From  the  expression  (in  Art.  86)  for  finding  the  num- 
ber of  permutations  of  n  things  taken  r  and  r  together,  we 
immediately  deduce  the  theorem  for  finding  the  number  of 
combinations  of  n  things  taken  in  the  same  manner.  For  the 
permutations  of  n  things  taken  two  and  two  together  being  n 
{n  —  \),  and  as  each  combination  admits  of  as  msiuj permutatiofis 
as  may  be  made  by  two  things  (which  is  2x1),  the  number 
of  combinations  must  be  equal  to  the  number  o^ permutations 
divided  by  2 ;  i.  e.  the  number  o^  combinations  o^n  things  taken 

two  and  two  together  is  — —- — -.     For  the  same  reason,  the 


124  ALGEBRA. 

combinations  of  n  things,  taken  three  and  three  together,  must 

71  (n-—l)  (n^2)  ,   .  -,      ,  ,  . 

be  equal  to  —     i    ^  \j \  ^^<^  ^^  general,  the  combma- 

tions  of  n  things  taken  r  ^d  r  together  must  be  equal  to 
n  (n-1)  (^-2)  .  .  .  {n-r+1) 

rT2".  S  ,  .  ,r 

Ex.  1.    Find  the  number  of  combinations  which  can  be 
formed  out  of  8  letters,  when  taken  5  at  a  time. 

_,,  .  8X7X6X5X4     _ 

Ihe  number  =- — - — - — - — -=56. 
1X2x3x4x5 

Ex.  2.  What  is  the  total  number  of  combinations  which  can 
be  formed  out  of  6  colours  taken  in  every  possible  way? 
No.  of  combinations  when  the  colours  are  taken — 

1  at  a  time     ----'---r=6 

2  -^-^  -15 


6.5.4 


=  20 
=  15 
=   6 


^  '    "    "         1.2.3.4.5.6     ■  ^ 


1.2.3 

6.5.4.3 

1.2.3.4 

6.5.4.3.2 

1.2.3.4.5 

6.5.4.3.2. 

_1 

Hence  the  total  number  =  63 


Ex.  3.  Eind  how  many  different  combinations  of  8  letters, 
taken  in  every  possible  way,  can  be  made. 

Ans.  255. 

*^*  Several  other  useful  and  interesting  subjects  of  an  elementary 
character  yet  remain  to  be  treated  of.  The  Editor  is  preparing 
for  publication  a  Second  Part,  which  will  embrace  these  subjects 


APPENDIX. 


GENERAL  PRINCIPLES,  PROPERTIES  OF  NUMBERS,  AND  OTHER 
EXPLANATIONS  IMPORTANT,  NOT  ONLY  FOR  THE  UNDERSTAND- 
ING OF  THE  CALCULATION,  BUT  FOR  SIMPLIFYING  THE  OPER- 
ATIONS   IN   AN    INFINITY    OF    WAYS. 


ON    THE    DIFFERENT    KINDS    OF    NUMBERS. 

1.  A  number  expresses  single  units,  or  parts  of  a  single 
unit,  or  at  the  same  time,  units  and  parts  of  units. 

By  unit  we  understand  a  whole  1,  and  by  a  part  of  a  unit, 
or  Si  fraction^  all  that  is  below  the  value  of  1. 

2.  The  number  which  expresses  units  only  is  called  a  whole^ 
a  simiyle^  or  a7i  incomplex  number.  That  which  expresses 
units  and  parts  of  units  taken  together,  is  called  ?i  fractionary^ 
a  compound^  or  a  complex  number.  That  which  expresses  on 
ly  parts  of  a  unit  is  called  diffraction, 

3.  The  number  which  does  not  specify  any  particular  kind 
of  units,  such  as  1,  2,  3,  4,  is  called  an  abstract  number. 
That  whose  kind  is  specified,  such  as  5  cents,  6  bushels,  is 
called  a  concrete  number. 

4.  The  number  terminated  by  2,  4,  6,  8,  or  0,  is  called  an 
even  number.  That  which  ends  with  1,  3,  5,  7,  or  9,  is  called 
an  odd  number, 

5.  The  number  which  has  no  exact  divisors,  in  whole  num- 
bers, but  itself  or  the  unit,  such  as  1,  2,  3,  5,  7,  11,  13,  &c., 
is  called  a  prime  number.  That  which  has  other  exact  di- 
visors than  itself  or  the  unit,  such  as  4,  6,  8,  9,  10,  12,  14,  15. 
&c.,  is  called  a  complex  number. 

11* 


126  APPENDIX. 

The  exact  divisors  of  a  complex  number  (or  multiple)  are 
called  submultiples.  Thus,  for  example,  1,  2,  3,  4,  6,  8,  and 
12,  are  submultiples  of  24. 

ON   THE   FOUR   RULES    OF   ARITHMETIC. 

6.  By  Addition  we  add  two  or  more  numbers  so  as  to 
make  but  one.     The  result  is  called  sum  or  total. 

By  Subtraction  we  cut  off  a  number  from  another  num- 
ber.    The  result  is  called -the  remainder^  surplus^  or  difference. 

By  Multiplication  we  take  a  number  called  multiplicand 
as  many  times  as  there  are  units  in  another  number  called  the 
multiplier.  The  result  is  named  the  product.  The  multipli- 
cand and  the  multiplier  are  called  the  two  factors  of  the 
product. 

By  Division*  we  ascertain  how  many  times  a  number  called 
the  divisor  is  contained  in  another  called  the  dividend.  The 
result  is  named  the  quotient.  The  dividend  and  the  divisor 
are  named  the  two  terms, 

7.  The  unit  neither  multiplies  nor  divides. 

8.  To  multiply  by  a  number  less  than  the  unit  is  to  di- 
minish the  number  multiplied  ;  hence  it  follows  that  to  multi- 
ply is  not  always  to  increase. 

To  divide  by  a  number  less  than  the  unit,  is  to  increase  the 
number  divided ;  hence  it  is  that  to  divide  is  not  always  to 
diminish. 

9.  Of  the  two  factors  of  a  product,  the  one  being  multi- 
plied an»t  the  other  divided  by  a  like  number,  the  result  re- 
mains the  same. 

10.  The  iwo  terms  of  a  divisor  being  multiplied  or  divided 
by  a  like  number,  the  quotient  remains  the  same. 

1 1.  The  general  product  of  several  numbers  is  always  the 
same  in  whatever  order  they  are  multiplied. 

12.  A  quantity  multiplied  or  divided  by  a  number,  gives 
the  same  product  or  the  same  quotient  multiplied  or  divided 
successively  by  the  factors  of  that  nymber. 

13.  If  the  quotient  be  greater  than  the  unit,  the  dividend 
is  greater  than  the  divisor,  and  vice  versa.  If  it  be  the  unit, 
the  divisor  is  equal  to  the  dividend. 


APPENDIX.  127 

14.  Jf  the  product  of  two  factors  be  less  than  their  sum, 
it  is  because  that  one  of  them  is  necessarily  the  unit. 

15.  To  double^  treble^  quadruple,  centriple,  &c.,  a  number, 
is  to  multiply  it  by  2,  3,  4,  100,  &c. 

16.  A  quantity  multiplied  and  divided  in  turn  by  a  like 
number,  becomes  again  what  it  was  at  first.  In  this  case, 
therefore,  the  shortest  way  is  to  dispense  with  both  multiplying 
and  dividing. 

17.  The  product  of  two  numbers  divided  by  one  of 
them,  gives  the  other. 

18.  The  quotient  multiplied  by  the  divisor  gives  the 
dividend  ;  the  dividend  divided  by  the  quotient  gives  the 
divisor. 

ON  THE  TWO  TERMS  OF  A  FRACTION. 

19.  Every  fraction  is  composed  of  two  terms :  the  f.rst 
mentioned  is  called  the  numerator^  the  second  denominator. 
The  numerator  indicates  how  many  equal  parts  of  the  unit 
are  contained  in  the  fraction :  the  denominator  gives  the  name 
of  those  parts. 

These  two  terms  of  a  fraction  are  assimilated  to  the  two 
terms  of  a  division.  The  numerator  represents  the  dividend  ; 
the  denominator  the  divisor. 

20.  If  the  numerator  be  equal  to  the  denominator  the  frac- 
tion is  equal  to  1.  If  the  numerator  is  less  than  the  denom- 
inator, the  fraction  is  less  than  1.  If  the  numerator  is 
greater  than  the  denominator,  the  fraction  is  greater  than  1. 

21.  Of  two  fractions  having  the  same  denominator,  the 
greatest  is  that  which  has  the  greatest  numerator.  Of  two 
fractions  having  the  same  numerator,  the  greatest  is  that 
which  has  the  smallest  denominator. 

22.  To  render  a  fraction  greater,  the  numerator  is  multi 
plied  without  touching  the  denominator,  or  the  denominator  is 
divided  without  touching  the  numerator: 

To  render  a  fraction  smaller,  the  numerator  is  divided  with- 
out touching  the  denominator,  or  the  denominator  is  multi- 
plied without  touching  the  numerator. 

23.  The  two  terms  of  a  fraction  being  multiplied  or  divi- 
ded by  a  like  number,  its  value  remams  the  same. 


128  APPENDIX. 

24.  Any  whole  number  may  always  be  put  indifferently 
under  the  form  of  a  fraction :  the  only  thing  is  to  give  it  the 
unit  for  denominator. 

25.  To  take  any  part  or  fraction  of  a  number,  is  to  multi- 
ply it  by  that  fraction. 

Thus  to  take  the  f  of  12=12xf=  V=Q- 

Then  if  the  fraction  to  take  has  the  unit  for  numerator,  we 
have  only  to  divide  by  the  denominator.  Thus  to  take  the 
■^,  ■^,  or  ^,  (fee,  of  a  number,  is  to  divide  tliat  number  by  2, 
3,  or  4. 

26.  To  increase  a  number  by  any  fraction  of  itself,  is  to 
multiply  it  by  a  new  fraction  whose  denominator  equals  the 
sum  of  the  two  gtven  terms,  and  whose  denominator  remains 
the  same. 

Thus  to  increase  60  by  j5^=60xi|  =  ^H^=^^- 

ON    RATIOS    AND    PROPORTIONS. 

27.  The  ratio  of  two  numbers  is  the  quotient  of  the  first 
number  di^dded  by  the  second.    Thus  the  ratio  of  15  to  5  is  3. 

The  connection  of  two  equal  ratios  is  called  the  geometrical 
proportion.  Thus  15  :  5  : :  6  : 2  is  one  proportion,  as  the  ratio 
of  15  to  5=3,  and  the  ratio  of  6  to  2= also  3. 

The  first  term  of  a  ratio  is  named  antecedent ;  and  the 
second  consequent. 

The  first  and  the  fourth  term  gf  a  proportion  are  called 
the  extremes  ;  the  second  and  third  are  called  the  mediums. 

The  mediums  may  always  exchange  places  without  dis- 
turbing the  proportion. 

28.  The  product  of  the  extremes  always  equals  that  of  the 
mediums. 

29.  We  determine  the  fourth  term  of  a  proportion  when 
unknown,  by  dividing  the  product  of  the  mediums  by  the  first 
term. 

ON    THE    SQUARES     OF    NUMBERS    AND    THEIR    ROOTS. 

The  square  number  or  second  power  of  a  number  is  that 
number  multiplied  once  by  itself. 

The  square  root  or  second  root  of  a  number  is  the  very 
number  which  has  been  raised  to  the  square. 


APPENDIX.  129 

This  is  the  natural  series  of  squares  up  to  100. 
Squares.     1  4  9  16  25  36  49  64  81   100. 
Roots.        12345     6789     10. 
Observe  that  the  successive  difference  between  the  squares 
always  exceeds  itself  by  two  unit#    Thus   from  1   to  4  the 
difference  is  3  ;  from  4  to  9  the  difference  is  5  ;  from  9  to  16 
the  difference  is  7,  &c. 

30.  A  number  which  is  not  a  perfect  square  is  called  a 
surd,  an  irrational  or  an  incommensurahle  number^  consequent- 
ly the  root  of  those  numbers  is  never  more  than  approx- 
imative. 

31.  To  increase  the  square  root  of  a  given  number  by  a 
unit,  add  to  that  number  the  double  of  its  root+1.    Example : 

Let  the  number  be  25  with  the  v/=5.  To  have  6  at  the 
root,  add  to  25  the  double  of  5  +  1  =  11,  and  you  will  have 
25 -f  11  =  36,  with  the  v/=6. 

If  the  root  of  the  given  number  have  a  remainder,  cut  it 
off  from  the  double-j-l  of  the  root,  the  new  remainder  will 
be  the  number  to  add  to  the  given  number.     Example  : 

Let  the  number  be  53,  of  which  the  v/=:74-the  remainder 
4.  To  have  8  at  the  root,  from  the  double  of  the  root  7+1 
=  15,  cut  off  the  remainder  4,  you  will  have  11.  Then  53  + 
11=64,  of  which  the  v/=8. 

32.  To  diminish  the  square  root  of  a  number  by  a  unit, 
take  from  that  number  the  double  of  its  root— 1.    Example  : 

If  the  number  be  64,  of  which  the  \/  =  8,  to  have  only  7  at 
the  root,  take  from  64  the  double  of  8  — 1  =  15,  you  willl^ave 
64-15=49,  of  which  the  x/=^l. 

If  the  root  of  the  given  number  have  a  remainder,  add  it 
to  the  double  — 1  of  the  root,  and  take  the  total  from  the 
given  number.     Example  : 

Suppose  the  number  is  86,  of  which  the  \/=9  +  the  remain- 
der 5.  In  order  to  have  the  root  =  8,  to  the  double  of  the 
root  9  —  1  =  17,  add  the  remainder  5,  you  will  have  17  +  5  = 
22.     Then  96—22=64,  of  which  the  v/=8. 

33.  The  proof  of  the  extraction  of  the  root  of  a  number 
is  done  by  multiplying  that  root  by  itself  and  joining  to  the 
product  the  remainder,  if  there  be  one ;  the  total  ought  to 
produce  again  the  number  whose  root  has  been  extracted. 

34.  Knowing  the  difference  of  two  numbers  and  that   of 


130  APPENDIX 

their  squares,  the  quotient  of  the  second  difference  divided  by 
the  first,  gives  the  sum  of  the  two  numbers,  which  enables  us 
to  determine  each  of  them  immediately. 

ON    THE    FACTORS    ANdS^UBMULTIPLES    OF    A    NUMBER. 

35.  Every  number  has  at  least  two  factors  ;  it  may  have 
more.  If  it  have  but  two  factors,  it  is  necessarily  a  prime 
number,  (5,)  and  those  two  factors,  consequently,  are  that 
number  itself  and  the  unit. 

36.  To  reproduce  by  multiplication  a  number  which  has 
more  than  two  factors,  the  greater  factor  must  be  multiplied 
by  the  lesser ; 

Or  the  factor  immediately  inferior  to  the  greater  by  the 
factor  immediately  superior  to  the  lesser ;  so  on,  all  through, 
always  following  the  same  order.     Example  : 

Suppose  the  number  be  24,  which  has  the  eight  factors,  1,  2, 
3,  4,  6, 8, 12,  and  24  ;  to  reproduce  that  number  we  shall  have  : 

24X1=:24,  or  12x2^=24,  or  8x3=24,  or  6x4=24. 

In  this  example  the  quantity  of  the  factors  is  even:  if  it  be 
odd^  then  it  is  the  medium  factor  which,  multiplied  by  itself, 
produces  the  number  in  question.     Example  : 

Suppose  the  number  be  64,  which  has  the  seven  factors,  1, 
2,  4,  8,  16,  32,  64 ;  to  reproduce  that  number  we  shall  have: 

64Xl==64,  or  34x2=64,  or  16x4=64,  or  8x8  =  64. 

From  this  article  we  shall  deduce  the  three  following  prin- 
ciples : 

37.  The  greatest  factor  of  a^ number  is  that  number  itself 
and  its  least  factor  is  the  unit. 

38.  The  greatest  factor  of  an  even  number  (that  number 
itself  excepted)  is  always  the  half  of  that  number,  and  its 
least  factor  (the  unit  excepted)  is  always  2. 

39.  Every  number  which  has  an  uyieven  quantity  of  fao 
tors  is  a  square  number  whose  root  is  the  mean  factor. 

40.  Observe  that  every  number  has  always  a  suhmultiple 
less  than  it  hdiS  factors.  Indeed  a  number,  whatsoever  it  be, 
figures  itself  amongst  its  factors,  but  never  amongst  its  sub- 
multiples. 

Observe,  further,  that  to  produce  a  number  by  the  multi- 
plication of  two  of  its  sub  multiples,  the  unit  will  never  be 


APPENDIX.  ISl 

one  of  those  submultiples,  seeing  that  it  cannot  in  any  way 
contribute  to  that  result. 

Thus  then,  excluding  the  unit  f7om  the  submultiples  of  a 
number,  we  shall  deduce  that  the  principles  established,  Arti- 
cles 36,  38,  and  39,  regarding  tb^  factors  of  a  number,  apply 
without  exception  to  its  submultiples. 

ON    ODD    AND     EVEN    NUMBERS. 

41.  The  sum  of  two  even  numbers  is  an  even  number. — 
That  of  two  odd  numbers  is  also  an  even  number.  That  of 
an  even  number  and  an  odd  number  is  an  odd  number. 

42.  The  difference  between  two  even  numbers  is  an  even 
number.  That  between  two  odd  numbers  is  also  an  even 
number.  That  betw  een  an  even  number  and  an  odd  number 
is  an  odd  number. 

43.  The  product  of  two  even  numbers  is  an  even  number. 
That  of  two  odd  numbers  is  an  odd  number.  That  of  an 
even  number  and  an  odd  number  is  an  even  number. 

44.  Every  odd  number  has  only  odd  factors  and  submul- 
tiples. 

45.  Every  prime  number,  with  the  single  exception  of  the 
number  2,  is  an  odd  number. 

ON    PROGRESSIONS. 

46.  The  arithmetical  progression  is  a  series  of  terms  suc- 
cessively increased  or  diminished  by  a  like  quantity.  In  the 
former  case  the  progression  is  called  increasing,  and  in  the 
latter  decreasing.  The  difference  between  the  terms  is  called 
2yroportion. 

The  geometrical  progression  is  a  series  of  terms  successive- 
ly multiplied  or  divided  by  a  like  quantity.  In  the  former 
case  the  progression  is  said  to  be  increasing,  and  in  the  latter 
deci-easing .  The  quantity  which  multiplies  or  divides  is  called 
p)roportion. 

There  are  five  quantities  to  be  considered  in  progressions  ; 
the  first  term,  the  last  term,  the  number  of  terms,  the  sum  of 
the  terms,  and  the  proportion  or  ratio. 

47.  The  last  term  of  an  arithmetical  progression  is  com- 


132  APPENDIX. 

posed  of  the  first  term,  more  as  many  times  the  ratio  as  there 
are  terms  before  it. 

Example. — Suppose  —2,  5,  8,  11,  14,  17,  of  which  the  ratio 
is  3.  Remark  that  the  last  term  17,  which  has^i'e  terms  be- 
fore it,  is  composed  of  the  fet  term  2+ (the  proportion  3x 
5)  =  17. 

48.  The  sum  of  the  terms  of  an  arithmetical  progression 
is  composed  of  the  sum  of  the  first  and  last  terms,  multiplied 
by  half  the  number  of  terms. 

Example, — Suppose  -^5,  7,  9,  11,  13,  15,  of  which  the  half 
of  the  number  of  terms  =r 3.  Then  the  first  terms  5 -{-the 
last  15=20  w^hichx3=:60^  the  sum  of  the  six  terms. 

49.  In  all  arithmetical  progre||ions  the  sum  of  the  first 
and  last  terms  equals  that  of  the  second  and  last  but  one,  or 
that  of  the  third  and  the  antepenultima,  and  so  on  all  through, 
still  observing  the  same  order. 

Example.— ^Vi^^o^Q  ~4,  7,  10,  13,  If^,  19,  22,  25.  Remark 
that  the  1st  and  the  8th  term =4 +25 =29  ; 
that  the  2d  and  the  7th      "     =7+22=29 ; 
that  the  3d  and  the  6th      "  =10+ 19=29  ; 
that  the  4th  and  the  5th     "  =13  +  16=29. 
In  the  example  just  given  the  number  of  terms  x^even.     If 
it  be  odd^  it  is  then  the  double  of  the  iniddle  term,  w^hich  equals 
th^  sum  of  the  other  taken  two  by  two  as  above. 
Example. — Suppose  -f-11,  16,  21,  36,  31.     Remark 
that  the  1st  and  the  5th  term  =  ll+31=42; 
that  the  2d  and  the  4th       "     =16  +  26=42; 
that  the  3d  doubled  =21  +21  =42.* 

50.  If  the  number  of  the  terms  of  an  arithmetical  pro- 
gression be  odd^  the  middle  t-erm  always  equals  the  sum  divi« 
ded  by  the  number  of  the  terms,  that  is  to  say,  that  if  that 
number  be  3,  or  5,  or  7,  &c.,  the  middle  term=^  or  \  or  i 
&c.,  of  the  sum  of  the  terms. 

Example, — Suppose  -f-10,  20,  30,  40,  50,  of  which  the  sum 
=  150. 

Then  that  sum  divided  by  5,  the  number  qjf  the  terms 
=  i|o=30. 

Hence,  as  we  see,  the  naiddle  term  =:  30. 

If  the  number  of  the  terms  be  even,  the  two  means  taken 


APPENDIX.  133 

together,  equal  the  sum  divided  by  half  the  number  of  the 
terms. 

Example. — Suppose  -f-5,  10,  15,  20,  25,  30,  of  which  the 
'uni  =  105.  Now  105  divided  bv  3,  the  half  the  number  of 
terms=r^§^i=35.  Hence  the  mo  middle  terms=15  +  20 
=35. 

51.  The  last  term  of  a  geometrical  progression  equals  the 
first,  multiplied  by  the  ratio  raised  to  a  power  of  a  degree 
equal  to  the  number  of  the  terms — 1. 

Exam-pie, — Suppose  -f^  3  :  6  :  12 :  24 :  48,  of  which  the  ratio 
is  2.  Now  the  progression  having -^vg  terms,  the  ratio  2 
raised  to  the  fourth  power  wdll  give  us  2x2x2x2=16. 
Then,  observe  that  the  last  ^erm= the  first  term  3x16=48. 

.  52.  To  find  out  the  sum  of  all  the  terms  of  a  geometrical 
progression,  the  last  term  is  multiplied  by  the  ratio  :  the  first 
term  is  taken  from  the  product,  and  the  remainder  is  divided 
by  the  ratio,  lessened  by  a  unit. 

Example. — Suppose  : :  2  :-6  :  18  :  54  :  162,  of  which  the 
ratio  is  3,  and  the  sum  242.  Hence,  the  last  term  162x3 
=486.  From  486  subtract  the  first  term  2,  remainder  484. 
Now  this  remainder  484  :  (the  ratio  3  — 1)  =  '*|-^=242,  sum 
of  the  terms. 

Remark. — All  these  difi'erent  principles  on  progression  ap- 
ply textually  to  increasing  progressions,  but  to  make  them 
applicable  to  decreasing  progressions,  it  is  only  necessary  to 
change  the  words  ^rs^J  into  last  and  last  into  Jirst. 

ON    DIVISIBLE    NUMBERS    WITHOUT    A    REMAINDER. 

53.  It  is  important,  for  abridging  the  calculation  in  a  mul 
titude  of  cases,  to  know  the  characters  which  render  one  num- 
ber exactly  divisible  by  another.  Thus,  we  may  always  di- 
vide without  any  remainder  : 

By  2,  all  even  numbers  without  exception. 

By  3,  all  even  or  odd  numbers  the  sum  of  whose  figures, 
considered  as  simple  units,  is  3  or  a  multiple  of  3. 

By  4,  all  even  numbers  whereof  the  two  last  figures  on  the 
right  are  divisible  by  4. 

By  5,  all  numbers  terminated  by  a  5  or  an  0. 

By  6,  all  even  numbers  already  divisible  by  3. 
12 


134:  APPENDIX. 

By  8,  all  even  numbers  whereof  the  three  last  figures  on 
the  right  are  divisible  by  8. 

By  9,  all  even  or  uneven  numbers  the  sum  of  whose  figures 
is  9  or  a  multiple  of  9. 

By  10,  100,  1000,  &c.,  \^\  numbers  ending  with  0,  00, 
000,  &c. 

By  11,  all  even  or  uneven  numbers  whereof  the  sum  of  the 
1st,  3d,  5th,  7th  figures,  &c.,  is  equal  to  the  sum  of  the  2d, 
4th,  6th,  8th,  (fee,  or  whose  diflTerenceis  11  or  a  multiple  of  11. 

By  12,  all  even  numbers  already  divisible  by  3  and  4. 

By  25,  all  even  or  uneven  numbers  whereof  the  two  last 
figures  on  the  right  are  divisible  by  25. 

PROPERTIES    AND    VARIOUS     EXPLANATIONS. 

54.  Of  two  unequal  numbers  the  greatest  equal  (the  sum 
4-the  difference)  divided  by  2,  and  the  smallest  equal  (the 
sum —ih.Q  diiference)  divided  also  by  2 ;  whence  it  follows 
that  the  sum  increased  by  the  difference  equals  twice  the 
greater  number,  and  that,  diminished  by  the  difference,  it 
equals  tiuice  the  smaller. 

55.  To  equal  two  unequal  numbers,  without  altering  their 
sum^  the  greater  is  diminished  by  half  the  difference,  and  the 
lesser  increased  by  the  other  half. 

56.  The  difference  between  two  numbers  cannot  be  superior 
nor  even  equal  to  the  greater  number,  but  it  may  equal  or 
exceed  the  lesser. 

57.  The  least  difference  possible  between  two  whole  num- 
bers, even,  or  between  two  whole  numbers,  odd,  is  2, 

58.  Knowing  how  many  times  A  is  greater  than  B,  we  at 
once  perceive  how  many  times  B  is  smaller  than  A,  preserving 
the  same  numerator  to  the  given  fraction,  and  forming  a  new 
denominator  from  the  sum  of  the  two  terms.     JExamples : 

A  being  >  B  by  i,  B  is  <  A  by  J. 
A  being  >  B  by  J,  B  is  <  A  by  /j. 

59.  Knowing  how  many  times  A  is  smaller  than  B,  we  in- 
stantly determine  how  many  times  B  is  greater  than  A,  pre- 
serving the  same  numerator  to  the  given  fraction,  and  forming 
a  new  denominator  of  the  given  denominator,  from  which  tho 
numerator  is  subtracted.     Examples  : 


APPENDIX.  135 

Abeing<Bby  I,  Bis>  Abyi. 
A  being  <  B  by  I,  B  is  >  A  by  |. 

60.  Knowing  that  the  number  A  is  equal  to  such  a  part  of 
the  number  B,  we  instantly  detennine  what  part  of  the  num- 
ber A  is  equal  to  the  number  B,  by  reversing  the  two  terms 
of  the  given  fraction.     Examples : 

A  being  I  of  B,  B=:f  of  A. 
A  being  I  of  B,  B=:f  of  A. 

61.  Knowing  how  many  times  the  sum  of  two  numbers 
contains  their  difference,  we  determine  the  proportion  of  one 
number  to  the  other  in  the  following  way  : 

Place  that  number  of  times  under  the  form  of  a  fraction, 
and  with  that  first  fraction  form  a  second,  whose  numerator  is 
equal  to  the  numerator  of  the  first,  less  its  denominator,  and 
w^hose  denominator  is  equal  to  the  numerator  of  the  first, 
more  its  denominator.  This  second  fraction  will  express  the 
proportion  of  the  lesser  number  to  the  greater,  and  reversed 
it  will  express  the  proportion  of  the  greater  to  the  lesser. 

Example, — Take  the  numbers  5  and  7,, of  which  the  sum  = 
12  and  the  difference  2 ;  the  one  contains,  therefore,  6  times 
the  other.  Then  6  in  a  fraction  [24]  =  J,  and  forming  our 
second  fraction  such  as  it  is,  we  shall  have  |-,  that  is  to  say 
that  the  lesser  number  ir:|-  of  the  greater. 

Another  example. — Take  the  numbers  60  and  35,  of  which 
the  sum =95  and  the  difference  25  ;  the  one  contains,  there- 
fore, 3  times  |-  the  other.  Then  3-|  =  y,  which,  according  to 
what  is  prescribed,  will  give  for  second  fraction  -^=xV  •  ^^ 
fine,  45  =  j^  of  60. 

62.  Every  number  having  two  unequal  figures,  lohen  read 
backwards^  differs  from  what  it  is  by  9  or  a  multiple  of  9.  If 
it  differ  by  9,  the  difference  between  the  two  figures  is  1.  If 
it  differ  by  twice  9,  by  three  times  9,  &c.,  the  difference  between 
the  two  figures  is  2,  3,  &c.  Thus  the  number  81,  read  back- 
wards, =18  ;  from  81  to  18  the  difference =63.  Now  63= 
seven  times  9.     So  the  difference  of  the  figures  8  and  1=7. 

63."  The  smaller  the  difference  is  between  two  numbers 

forming  a  like  sum,  the  greater  their  product  is.  Examples : 

riX7=   7  riX8=  8 

No.  1.       12x6  =  12       No.  2.      J  2x7  =  14 

Sum  8.       I  3X5  =  15       Sum  9.      13x6  =  18 

[4x4=16  [4X5=20 


J.36  ,     APPENDIX. 

Now,  the  two  examples  here  given  furnish  us  with  the  fol- 
lowing remarks  : 

1st.  The  one  of  the  two  numbers  which,  multiplied  one  by  the 
other,  give  the  smallest  product,  is  always  the  unH,  whether 
the  sum  be  odd  or  eveii  ;  '     ^> 

2d.  If  the  sum  be  even^  No.  1,  each  of  the  two  numbers 
which  give  the  greater  product  is  equal  to  half  the  sum ; 

3d.  If  the  sum  be  odd^  No.  2,  the  two  numbers  which  give 
the  greater  product  differ  between  themselves'  by  the  unit. 

64.  There  are  some  quantities  which,  according  to  the 
nature  of  the  question,  can  only  be  fractionary.  Thus,  for 
instance,  when,  in  speaking  of  workmen^  birds^  ^99^-,  <^c.,  we 
mention  the  half^  thirds^  quarters^  &;c.,  it  is  necessarily  sup- 
posed that  those  quantities  are  exactly  divisible  by  2,  3,  4,  &;c. 

But,  to  know,  with  accuracy,  the  number  whereby  a  quan- 
tity of  this  kind  becomes  divisible,  if  increased  by  one  of  its 
parts,  add  the  two  terms  of  the  fraction  which  expresses  that 
part,  the  sum  will  be  the  answer. 

Thus,  for  example,  if  1  increase  by  \  the  contents  of  a  bas- 
ket of  eggs,  I  conclude  that  those  contents,  at  first  exactly  di- 
visible by  7,  is  now  divisible  by  7 -|- 5  =  A.  12. 

If,  on  the  contrary,  the  quantity  of  one  of  its  parts  be  di- 
minished, you  will  determine  the  number  by  which  it  becomes 
divisible,  taking  the  numerator  from  the  denominator.  The 
remainder  will  be  the  answer. 

Thus,  for  example,  if  I  diminish  by  ^  the  birds  of  an  aviary, 
I  conclude  that  their  number,  at  first  exactly  divisible  by  7,  is 
now  divisible  by  7— 5= A.  2. 

65.  This  is  an  operation  which  often  occurs  in  my  solutions. 
In  order  that  the  reader  may  perfectly  understand  it,  I  am 
about  to  give  here  an  explanatory  example. 

Let  us  suppose  that  the  question  is  to  divide  (.t+4)  into 
two' parts,  one  of  which  =:  the  |  of  the  other.  Make  the  sum 
of  the  two  terms  of  the  given  fraction,  you  will  have  3-f-5  = 
8,  which  indicates  that  the  lesser  part  ought  to  have  the  \  of 
the  number  (aT+4)  and  the  greater  the  |-. 

Operation  :      (a? +4)  f = j — - —  the  lesser  number. 

(a? +4)  -1=   — ^ —   the  greater  numbeF. 


APPENDIX.  137 


MISCELLANEOUS  PROBLEMS. 

66.  Divide  46  into  two  such  ^arts  that  the  sum  of  the  quo- 
tients obtained  after  dividing  one  by  7  and  the  other  by  3 
may  be  10.  Ans.  28  and  18. 

67.  Divide  $1170  between  three  persons,  A,  B,  C,  propor- 
tionally to  their  ages :  B's  age  is  one  third  greater  than  that 
of  A,  who  is  but  half  of  C's.    What  is  the  share  of  each  1 

Ans.  A,  1270  ;    B,  $360  ;    C,  $540. 

68.  A  capital  is  such,  that,  augmented  its  simple  interest 
for  5  years  at  4  per  cent,,  it  raises  to  the  amount  of  $8208. 
What  is  the  capital  ?  Ans,  $6840. 

69.  A  capitalist  placed  the  |  of  his  stock  at  4  per  cent, 
and  the  remaining  J  at  5  per  cent. :  it  produces  in  all  $2940. 
What  was  the  whole  sum  lent  ?  Ans,  $70,000. 

70.  If  two  sums  of  money  be  placed  at  interest,  one  of 
$5500  at  4  per  cent.,  and,  4|-  years  afterwards,  $8000  be 
placed  at  5  per  cent.,  in  what  time  will  the  sums  produce  the 
same  interest  1 

Ans,  10  years  from  the  time  the  first  sum  was  placed. 

71.  1  had  a  certain  sum  in  my  purse  :  I  took  out  the  third 
of  its  contents :  I  then  put  in  $50 ;  some  time  afterwards  I 
took  out  the  fourth  of  what  it  then  contained,  and  put  in  $70 
more,  after  which  it  contained  $120.  How  'many  were  in  ii 
first  ?  Ans.  $25. 

72.  Says  A  to  B,  Give  me  $100,  and  we  shall  have  equal 
sums.  Give- me,  says  B  to  A,  $100,  and  I  shall  have  double 
what  thou  hast.     How  many  had  each?  Ans,  $500  and  $700. 

73.  Find  two  numbers  whose  difl?erence,  sum,  and  product, 
may  be  to  one  another  as  the  numbers  2,  3,  and  5. 

Ans,  2  and  10. 

74.  The  sum  of  two  numbers  is  13  and  the "  difference  of 
their  squares  is  39.     What  are  the  numbers  ?     Ans.  5  and  8. 

75.  A  and  B  together  have  but  the  f  of  C ;  B  and  C  to- 
gether 6  times  A,  and  if  B  had  $680  more  than  he  really  has, 
he  would  have  as  much  as  A  and  C  together.  How  many  dol- 
lars has  each?  Ans,  A,  $200  ;  B,  $360  ;  C,      * 

12* 


138  APPENDIX. 

76.  What  is  that  number  whose  seventh  part  multiplied  by 
its  eighth  and  the  product  divided  by  3  would  give  298 1  for 
result  ?  Ans.  224. 

77.  What  are  two  numbers  whose  product  is  750  and 
whose  quotient  is  3^  ?  <i  Ans.  50  and  15. 

78.  A  person  being  questioned  about  his  age,  replied  :  My 
mother  was  twenty  years  of  age  at  my  birth,  and  the  number 
of  her  years  multiplied  by  mine  exceed  by  2500  years  her 
age  and  mine  united.     What  is  his  age  ?  Ans.  42. 

79.  A  gentleman  bought  some  furniture  and  sold  it  shortly 
after  for  $144;  by  which  he  gained  as  much  per  cent,  as  it 
cost.     Required  the  first  cost.  Ans.  $80. 

80.  Determine  two  numbers  whose  sum  shall  be  41,  and 
the  sum  of  whose  squares  shall  be  901.       Ans.  15  and  26. 

81.  The  difference  between  two  numbers  is  8,  and  the  sum 
of  their  squares  544.     What  are  the  numbers  ? 

A71S.  12  and  20. 

82.  The  product  of  two  numbers  is  255.  and  the  sum  of 
their  squares  514.  What  are  the  numbers?     Ans.  15  and  17. 

83.  Divide  the  number  16  into  two  such  parts,  that  if  to 
their  product  the  sum  of  their  squares  be  added,  the  result 
will  be  208.  Ans.  4  and  12. 

84.  What  is  the  number  which  added  to  its  square  root  the 
sum  shall  be  1332?  Arts.  1296. 

85.  What  is«the  number  that  exceeds  its  square  root  by 
48 J?  Ans.  56^. 

86.  Find  two  numbers,  such  that  their  sum,  their  product, 
and  the  difference  of  their  squares,  may  be  equal  ? 

Ans.    3+^5     1±^5 
2      '        2     * 

87.  Find  two  numbers,  whose  difference  multiplied  by  the 
difference  of  their  squares  gives  for  product  160,  and  whose 
sum  multiplied  by  the  sum  of  their  squares  gives  580  for  pro- 
duct. Ans.  7  and  3. 

88.  What  is  the  ratio  of  a  progression  by  difference  of  22 
terms,  the  first  of  which  is  1  and  the  last  15?  Ans.  f. 

89.  There  is  a  number  of  two  figures,  such,  that  if  you  di- 
.  vide  it  by  the  sum  of  its  figures,  then  inverting  the  number 


APPENDIX.  139 

and  dividing  this  new  number  by  the  sum  of  its  figures,  the 
difference  of  the  two  quotients  is  equal  to  the  difference  of  its 
figures,  and  the  product  of  the  two  quotients  to  the  number 
itself.     What  is  the  number  ?  Ans.  IS. 

90.  Triple  Louisa's  age  is  as  ngpch  above  40  as  the  third  of 
her  age  is  under  10.     What  is  her  age?  Ans,  15. 

91.  Being  questioned  how  many  eggs  I  had  put  in  an  ome- 
let, I  answered  that  the  |  of  the  whole  augmented  by  the  |- 
of  an  egg  exceeded  the  f  of  the  whole  by  the  square  root  of 
all  the  eggs  employed.  A^is,  16. 

92.  Fortune  was  against  me  this  morning,  but  it  favoured 
me  this  evening  :  I  doubled  the  remainder  of  my^oney,  and 
instead  of  £5  that  I  lost,  I  have  now  £10  gain.  How 
many  pounds  have  I  now  ?  Ans,  £30. 

93.  If  my  salary  were  doubled,  said  a  comedian,  it  would 
be  96  shillings  more  than  the  square  of  the  twenty-fifth  part 
of  what  it  is.     What  was  if?  Ans.  1200  shillings. 

94.  There  are  two  numbers  whose  sum  is  63,  and  |-  of  one 
is  equal  to  quintuple  the  other.     What  are  they  ? 

Ans.  56  and  7. 

95.  Add  5  to  the  number  of  my  children,  double  the  re- 
sult, and  you  will  have  triple  my  family.  How  many  chil- 
dren have  n  Ans.  10. 

96.  Two  numbers  are  such,  that  triple  the  less  is  3  units 
more  than  the  greater ;  and  if  the  greater  be  augmented  its 
■A  and  the  less  its  f ,  the  former  becomes  double  the  latter. 
What  are  they  ?  Ans.  33  and  12. 

97.  A  brother  said  to  his  sister :  I  would  want  ^  of  your 
pounds  to  make  £30 ;  give  me  them.  I  would  want,  replied 
the  sister,  |  of  yours  to  have  £40 ;  will  you  give  me  them  ? 
How  many  pounds  had  each  1  Ans.  £25  and  £20. 

98.  By.^neans  of  a  legacy  that  quintupled  his  revenue, 
Michael  can  spend  $2  a  day  and  lay  up  yearly  the  |^  of  the 
legacy.     What  was  it?  Ans.  $800. 

99.  A  fruiterer  bought  pomegranates  at  5  cents  for  6,  and 
by  selling  them  at  3  cents  for  2  she  gained  $1.20.  Ho^ 
many  did  she  buy?  Ans.  180. 

100.  Eobert  buys  a  horse  and  sells  him  immediately.  If 
he  had  gained  ^  more,  his  gain  would  have  been  douljle,  and 


140  APPENDIX. 

would  have  been  but  10  shillings  less  than  half  the  first  cos^ 
Required  the  first  cost  and  selling  price. 

A71S,  100  and  120  shillings. 

101.  Determine  two  numbers  whose  diflference  shall  be 
equal  to  one  of  them,  and^  whose  product  shall  be  18  more 
thantriple  their  sum.  Ans.  12  and  6, 

102.  Of  three  numbers  the  mean  is  l  greater  than  the  less, 
and  the  former  is  ^  less  than  the  greater ;  now  if  each  was  re- 
duced i  their  sum  would  be  reduced  19  units.  What  are 
the  three  numbers  ?  Ajis.  24,  18,  and  15. 

103.  Three  times  a  number,  less  20,  is  as  much  above 
-double  the  same  number  as  its  fourth  part,  plus  2,  is  undei 
its  half.     What  is  the  number  ?  Ans.  24. 

104.  Louisa  buys  2^  lbs.  of  sugar,  at  6d.  per  lb.,  and  gives 
in  payment  a  piece  of  silver  such  that  the  square  of  the  piece 
returned  exceeds  the  triple  of  the  expense  by  a  sum  equal  to 
the  return.     Required  the  value  of  the  piece  given  hy  Louisa. 

,  Ans.  Is.  Sd. 

105.  By  what  number  should  3  be  multiplied  in  order  that 
the  ^2  of  the  product  may  be  equal  to  the  sum  of  the  two 
fkctors?  Ans.  By  12. 

106.  When  the  granddaughter  was  born,  the  grandfather 
wag  3|-  times  the  granddaughter's  present  age,  and  10  years 
after,  the  latter  was  but  -J  of  the  grandfather's  aforesaid  age. 
Required  their  respective  ages.  Ans.  90  and  20. 

107.  Determine  three  numbers,  of  which  the  greater,  equal 
to  the  sum  of  the  two  others,  is  also  equal  to  the  |  of  their 
product,  and  of  which  the  less  is  but  the  ^  of  the  other  two 
together.  Ans.  24,  18,  and  6. 

108.  An  uncle  claims  the  ^^  of  an  inheritance,  the  nephiew 
J»  and  the  niece  the  remainder.  The  product  of  the  parts  of 
the  two  latter  is  13  millions  less  than  the  square  of  the  un- 
cle's part.     What  was  the  inheritance  ?  Ans.  $12 fiOO. 

109.  I  drew  two  numbers,  whose    sum   is  equal  to  4  times 
their  difference,  and  the  greater  of  which,  plus  the  difference,  • 
exceeds  the  less  by  48  units.     What  are  the  numbers  ? 

A?is.  60  and  S^. 
llj}.  My  Fatch  is  very  methodical  in  its  time.     Were  I  to 


APPENDIX.  141 

let  it  go,  it  would  mark  the  exact  time  every  two  months  reg- 
ularly.    What  is  the  yariation  of  my  watch  per  hour  ? 

Ans.  "I  minute  fast  or  slow. 

111.  Four  years  ago,  the  sister  was  |  of  the  brother's  years 
older  than  the  brother :  four  year^^ence  the  brother  will  be 
J  of  the  sister's  years  less  than  the  sister.  What  is  the  age 
of  eachi  Ans^  Sister,  16;  brother,  14. 

112.  Had  I  double  my  gain,  said  a  gamester,  I  would  have 
squared  the  number  of  my  pounds ;  were  it  but  -J-  what  it  is, 
I  would  have  tripled  them  only.     What  was  the  gain  ? 

A71S.  £36. 

113.  My  uncle  spent  J  of  his  lifetime  bachelor,  ^  married, 
and  ^  a  widower.  When  he  married  my  aunt,  she  was  i  of 
his  years  younger  than  he,. and  8  years  after  his  years  were  J 
greater  than  hers.     At  what  age  did  each  die  ? 

Ans.  At  72  and  at  40. 

114.  I  have  one  sister  and  two  brothers,  the  two  latter  are 
twins  :  now  my  sister's  age  is  equal  to  the  sum  or  the  prod- 
uct, just  as  you  please,  of  my  brothers'  ages.  Please  find 
my  sister's  and  brothers'  ages  separately. 

Ans.  Sister's  age,  4 ;  twins,  2  years  each. 

115.  Subtract  ^  of  my  brother's  age,  and  ^  of  my  sister's 
age,  you  shall  have  made  them  twins  and  the  sum  of  their 
years  will  have  lessened  2.     What  is  the  age  of  each  ? 

A71S.  24  and  18. 

1 16.  A  gamester  being  questioned  about  his  gain,  answered : 
One  of  the  factors  of  my  gaih  is  but  half  the  other,  and  their 
sum  is  but  half  my  gain.     How  many  dollars  did  he  win  ? 

Ans.  18. 

117.  Find  two  numbers,  such,  that  the  square  of  the  lesser 
may  be  equal  to  |  of  the  greater,  and  the  greater,  diminished 
its  f ,  may  exceed  the  lesser  by  ^.  Ans.  20  and  4. 

118.  The  product  of  two  numbers  equals  three  times  their 
sum,  and  their  quotient  is  3.     What  are  the  numbers  ? 

«  Ans.  12  and  4. 

119.  Square  ^  plus  1  of  the  years  that  I  am  short  of  a 
quarter  of  a  century  and  you  will  produce  my  age.  What  is 
it?  Ans.  16. 

li^O.  A  gamester  lost  at  the  first  game  the  square  of  ^  of 


142  APPENDIX. 

the  dollars  he  had  about  him ;  but  at  the  second  round  he 
quintuples  his  remainder,  and  withdraws  neither  gainer  nor 
loser.     How  many  dollars  had  he  ?  Ans,  80. 

121.  I  am  going  to  add  %^new  shelves  to  my  library,  each 
of  which  will  hold  20  volumes  more  than  the  ten  already  ex- 
isting, and  so  I  shall  have  1000  volumes  in  all.  How  many 
have  I  now  ?  A71S.  600. 

122.  Multiply  half  the  father's  age  by  half  the  son's  age  and 
you  will  have  the  square  of  the  son's  age  ;  this  square  is 
equal  to  double  the  sum  of  their  ages.     How  old  is  each  ? 

Ans.  40  and  10. 

123.  The  breadth  of  my  room  is  but  the  f  of  its  length. — 
As  broad  as  it  is  long,  it  would  contain  144  square  feet  more. 
Required  its  dimensions.  Ans.  24  by  18. 

124.  The  difference  between  the  |-  of  my  age,  less  5,  and 
its  I,  plus  3,  is  the  square  root  of  my  age.     "What  is  it  1 

Ans.  36  years. 

125.  The  product  of  two  numbers  is  220.  If  from  the 
greater  you  subtract  the  difference,  their  product  will  lessen 
99  units.     Find  the  two  numbers  by  one  unknown  term. 

Ans.  20  and  11. 

126.  What  is  the  number  of  your  house  ?  The  sum  of  its 
digits,  considered  as  units,  is  equal  to  -^  of  the  number.  Find 
it.    '  Ans.  54. 

127.  The  square  of  the  difference  of  two  numbers  is  equal 
to  its  sum,  and  I  of  the  former  is  equal  to  ^  of  the  latter. 
What  are  the  two  numbers  1  Ans.  10  and  6. 

128.  An  officer  gave  the  following  indication  of  the  number 
of  his  regiment :  One  of  its  factors  is  to  the  other  : :  1:5, 
and  ,their  sum  is  to  their  product  : :  6  :  25.  What  was  the 
number?  Ans.  125. 

129.  My  age  is  composed  of  two  figures,  and  read  back- 
wards it  makes  me  I  older.     What  is  it  ?  Ans.  45. 

130.  The  product  of  two  ii umbers  is  J  more  than  their 
sum,  and  is  equal  to  triple  their  difference.     What  are  they  ? 

A71S.  2  and  6. 

131.  When  the  brother's  age  was  the  square  of  the  sister's, 
she  was  ^  of  fhe  brother's  present  age,  and  8  years  hence  me 


APPENDIX.  143 

sum  of  their  ages  will  be  augmented  its  |.     WhaC  is  the  age 
of  each?  Ans.  21  and  15. 

132.  My  garden,  longer  than  broad,  contains  900  square 
yards ;  if  with  the  same  superfici^  it  were  a  perfect  square, 
its  length  would  diminish  the  |,  how  much  should  its  breadth 
be  augmented  ?  Ans,  Its  f. 

133.  Eight  years  ago,  the  brother's  age  was  equal  to  his  two 
sisters'  ages  together ;  8  years  hence  it  will  be  but  the  f  of  it, 
and  at  the  same  period  the  age  of  the  youngest  is  exactly  the 
^  of  the  three  ages  united.     What  are  the  ages  ? 

Ans.  24,  20,  and  12. 

134.  Divide  a  basket  of  pears  among  three  sisters,  so  that 
the  part  of  the  older  may  be  to  that  of  the  second  ::  i  :  ^', 
and  that  of  the  second  to  the  part  of  the  youngest  : :  -g- :  i ; 
then  the  sum  of  their  squares  will  be  549.  How  many  pears 
had  each'?  Ans,  18,  12,  and  9. 

135.  A  gambler  questioned  about  his  gain,  replied  :  Divided 
by  its  least  sub  multiple,  it  diminishes  £10 ;  divided  by  its 
greatest  submultiple,  it  is  diminished  £12.  How  many 
pounds  did  he  gain  ?  Ans.  £15. 

136.  The  number  of  my  house  has  seven  sub-multiples  be- 
sides Unit.  The  fourth,  in  the  order  of  magnitude,  is  ^  of  the 
number.     What  is  it^  Afis.  36. 

137.  1  sold  ^  of  my  basket  of  eggs  in  one  house,  and  25 
in  another.  Triple  the  remainder,  and  you  shall  reproduce 
the  primitive  contents.     What  was  it  ?  Ans.  60  eggs. 

138.  I  have  in  my  right  hand  double  the  contents  I  have  in 
my  left ;  but  if  I  put  in  my  left  hand  the  square  root  of  the 
number  in  my  right,  each  will  contain  the  same  number.  How 
many  did  each  contain  ]  Ans.  16  and  8. 

139.  If  I  had  paid  ^  more  for  my  watch,  its  price  would 
have  been  less  by  £4  than  double  what  it  cost  me.  What. 
did  I  pay  for  it  ]  Ans.  £6. 

140.  One  of  the  factors  of  a  multiplication  is  5,  and  the- 
other  is  8  less  than  the  product.     What  is  the  other  factor!: 

Ans.  %. 

141.  A  gentleman  looking  at  his  watch,  was  asked  thetmie ;; 
he  reflected  an  instant  and  then  replied:  The  square  of  the 


144  APPENDIX. 

preseiit  time,  plus  its  root,  is  half  greater  than  less  its  root. 
What  o'clock  was  it  1  A71S.  5. 

142.  The  sum  of  the  four  terms  of  an  arithmetical  pro- 
gression  is  44  ;  that  of  the^wo  first,  18.  What  are  the  four 
terms?     •  Ans.  S,  10,  12,  M. 

143.  There  is  4  difference  between  two  numbers,  and  their 
sum  is  less  than  their  product.     Required  the  two  numbers. 

A71S.  2  and  6. 

144.  The  difference  of  two  numbers  equals  |-  of  the  greater, 
and  represents  the  square  of  the  less.  What  are  the  num- 
bers? Ans,  30  and  5. 

145.  There  are  two  unequal  numbers ;  the  less  is  equal  to 
I  of  their  sum,  and  their  sum  equals  3^  of  their  product. 

Ans.  6  and  4. 

146.  The  sum  of  two  numbers,  plus  their  difference,  makes 
100,  and  their  difference  joined  to  their  quotient  equals  45. 
What  are  the  numbers?  Ans.  50  and  10. 

147.  I  received  this  morning  a  basket  of  peaches :  I  laid 
■|-  by  for  myself,  and  the  remainder,  a  prime  number,  I  divided 
amongst  my  children  in  equal  parts.  How  many  children 
have  I?  Ans.  11. 

148.  The  f  of  the  numerator  equals  the  |-  of  the  denomi- 
nator, and  the  sum  of  the  two  terms  is  1 1  more  than  the  pro 
duct  of  ^  of  the  denominator  by  J  of  the  numerator.  What 
is  the  fraction?  Ans,  |. 

149.  I  bought  a  horse  yesterday  and  sold  him  at  a  profit 
equal  to  the  f,  less  £11,  of  my  outlay,  by  which  I  gained  20 
per  cent.     Required  the  first  cost  and  selling  price. 

Ans.  £20  and  £24. 

150.  The  four  terms  of  a  proportion  make,  together,  100. 
The  first  is  equal  to  the  third  and  the  ratio  is  4.  What  are 
die  terms  ?  •  .       Ans.  40  :  10  : :  40 :  10. 

151.  The  dividend  is  equal  to  the  square  of  the  divisor,  and 
their  sum  added  to  their  quotient,  is  equal  to  840.  What  is 
the  dividend?  what  is  the  divisor  ?  Ans.  784  and  28. 

152.  One  square  is  quadruple  another,  and  their  sum  added 
to  the  sum  of  the  two  roots  is  530.  What  are  the  two 
squares  ?  Ans.  400  and  100. 


APPENDIX.  145 

153.  A  milkmaid  sells  hens'  eggs  and  ducks'  eggs.  Their 
mean  price  is  16  cents  the  dozen.  Now  6  dozen  of  the  latter 
are  worth  10  dozen  of  the  former.  Required  the  price  of  each 
dozen.  ^    Afis.  12  and  20  cents. 

154.  What  cost  these  six  pounds  of  sugar?  If  they  were 
worth  3  cents  per  lb.  more,  my  outlay  would  have  been  J 
more.     What  did  it  cost  per  lb.  1  Ans.  15  cts. 

155.  The  four  terms  of  a  proportion  equal  100.  Add  1  to 
each  term  and  they  shall  be  still  proportional.  What  are 
they  ?  Ans.  25  :  25  : :  25  :  25. 

156.  The  sum  of  two  numbers  exceeds  their  difference  by 
12,  and  their  product  exceeds  their  sum  by  24.  What  are  the 
two  numbers?  Ans.  6  and  6. 

157.  A  number  squared  is  such,  that  if  25  be  taken  from 
it,  the  remainder  will  be  7^  times  its  root.  Required  the 
square  ?  Ans.  100. 

158.  An  annuity  placed  at  13 J  per  cent,  per  annum,  pro- 
duces monthly  a  rent  equal  to  its  square  root.  What  is  the 
annuity?  Ans.  $8100. 

159.  The  product  of  two  numbers  is  120.  Add  1  to  each, 
and  their  product  shall  be  150.     What  are  the  numbers'? 

Ans.  24  and  5. 

160.  Two  numbers  are  equal.  If  3  be  added  to  each,  their 
product  will  increase  51.     What  are  the  two  numbers? 

Ans.  7  and  7. 

161.  The  sum  and  difference  of  two  numbers  taken  together, 
is  24  ;  the  product  and  the  quotient,  51.  What  are  the  num- 
bers? Ans.  4  and  12. 

162.  The  number  of  my  years  is  divisible  by  5  and  7;  but 
the  quotient  by  7  is  the  lesser  by  4.     What  is  my  age  ? 

Ans.  70. 

163.  The  two  terms  of  a  division,  added  to  their  quotient, 
make  109.  Take  8  from  the  dividend,  and  the  quotient  w^ll 
be  less  by  2.     What  are  the  three  terms  ? 

Ans.  84,  4,  and  21. 

164.  The  square  of  the  numerator  is  1  more  than  the  de- 
nominator, and  the  sum  of  the  two  terms  is  1  more  than  twice 
their  difference.     What  is  the  fraction  ?  Ans.  f. 

13 


146'  APPENDIX. 

165.  The  two  terms  of  a  division  sum  stre  the  same  as  the 
two  terms  of  another,  but  in  an  inverted  order.  The  sum  of 
the  four  terms  is  80,  and  that  of  their  quotients  2^.  What 
are  the  terms  of  these  division  sums?         Ans.  24  and  16. 

166.  Half  the  dividend  is  equal  to  the  square  of  the  divi- 
sor; half  the  divisor  equals  the  square  of  the  quotient.  Ke- 
quired  the  dividend  and  divisor.  Ans.  128  and  8. 

167.  Two  numbers  are  : :  8  :  5.  Subtract  .5  from  the  first 
to  add  to  the  second,  and  they  will  be  : :  7  :  6.  What  are  the 
two  numbers  ?  Ans.  40  and  25. 

]  68.  How  many  miles  from  A  to  B  ?  said  an  inquisitive  per 
son.  He  received  for  answer  that  their  number  had  but  -two 
factors,  whose  sum  was  20.     What  is  that  number? 

Ans.  19. 

169.  The  product  of  the  two  terms  of  a  fraction  is  120. — 
Add  1  to  the  numerator,  subtract  1  fi'om  the  denominator,  their 
quotient  will  be  1.     What  is  the  fraction?  Ans.^. 

170.  The  failure  of  an  insolvent  debtor  took  away  the  |  of 
the  capital  that  I  had  placed  in  his  hands.  The  interest  of  the 
remainder  placed  at  5  per  cent,  is  equal  to  the  square  root  of 
the  first  capital.     What  was  it  ?  A^is^.  $10,000. 

171.  If  you  find  me  too  old,  read  my  age  backwards,  and 
you  will  make  me  the  |-  younger.  What  was  the  person's 
age?  Ans.  81. 

172.  Find  me  three  numbers  in  arithmetical  progression, 
the  third  of  which  shall  be  equal  to  the  square  of  the  first, 
and  the  second  triple  the  ratio.  Ans.  2,  3,  4. 

173.  I  travelled  6  miles  an  hour,  said  a  pedestrian  :  had  I 
travelled  7|-,  I  should  have  arrived  8  hours  sooner.     Required 

the  length  of  the  journey.  Ans.  240  miles. 

174.  If  you  double  the  denominator  of  a  fraction,  the  sum 
of  its  two  terms  will  be  22,  and  if  you  triple  its  numerator, 
the  sum  will  only  be  21.     Determine  the  fraction.    Ans.  J. 

175.  Increase  the  contents  of  my  purse  £3,  and  it  will  be- 
come a  perfect  square.  Jf,  on  the  contrary,  you  lessen  it  £3, 
it  will  contain  but  the  root  of  the  aforesaid  square.  What  are 
the  contents?  Ans.  £6. 

176.  A  lady  questioned  about  hei-  age,  answered  :  Increase 


APPENDIX.  147 

it  the  f ,  and  in  that  state  lessen  it  the  f ,  and  you  shall  have 
made  me  25  years  younger.     What  was  the  lady's  age  ? 

A71S.  60. 

177.  The  figure  9  is  not  to  be  found  among  those  that  rep- 
resent my  wife's  age  ;  but  if  yo^read  backwards  the  two 
figures  that  do,  she  will  be,  to  her  great  displeasure,  older  by 
a  number  of  years  between  55  and  70.  Please  tell  me  my 
wife's  age.  Ans,  18 

178.  Well,  Tom.,  what's  the  time  ?  Square  the  §  of  it  and 
you  shall  have  the  hour  that  shall  strike  6  hours  hence.  What 
o'clock  was  it?  Ans.  10  o'clock. 

179.  How  many  children  have  you,  sir?  Their  number  is 
equal  to  y ;  the  square  of  y  is  equal  to  2y '  plus  9x,  and  y  ex- 
ceeds X  the  I".  Find  the  number  of  my  childi»en  with  one 
equation,  only  and  one  unknown  term.  A7is,  9. 

180.  Find  two  numbers  whose  sum  :  their  difference  : :  7:3, 
and  whose  product  increased  10  is  equal  to  the  square  of  f  of 
the  greater  number.  Ans.  15  and  6. 

181.  Part  of  my  capital  is  placed  at  5  per  cent.,  the  remainder 
at  6 ;  their  sum  is  $35,000.  I  of  the  revenue  of  the  first 
part  is  equal  to  -^  that  of  the  second.     Required  the  two  parts. 

Ans.  $15,000  and  $20,000. 

182.  Four  numbers  are  in  geometrical  proportion;  the  ra- 
tio is  ^.  The  fourth  term  is  square  of  the  first,  the  quotient 
of  the  second  by  the  third  is  1.     Bequired  the  four  terms. 

Ans.  4:  8  ::  8  :  16. 

183.  The  number  x  exceeds  the  number  y  by  the  whole 
root  of  X,  and  ^  of  x  equals  the  ^q  of  y.  Determine  the  two 
numbers  with  one  unknown  term.  Ans.  36  and  30. 

184.  Yesterday  I  bought  5  yds.  of  black  cloth  and  6  yds. 
of  blue,  ^his  morning  I  bought  3  yds.  of  blue  and  9  of  black, 
and  my  expense  was  the  same  as  that  of  yesterday.  Their 
mean  price  was  85  shillings.  Required  the  price  of  a  yard  of 
each.  Ans.  30  an(^0  shillings. 

185.  Six  years  hence,  said  a  lady,  my  daughter's  age  will 
be  the  square  of  what  it  was  6  years  ago.  \^hat  is  her  age 
to-day?  Ans.  10  years. 

186.  The  sum  of  two  numbers  is  4  times  their  difference, 


148  APPENDIX. 

and  the  difference  5^^  of  their  product.    What  are  the  two  num- 
bers? Afis.  10  and  6. 

187.  An  officer  being  questioned  about  the  number  of  men 
in  his  detachment,  answered  :  Their  number  has  but  3  factors, 
whose  sum  is  31.     What  ^fas  their  number?         Ans,  25. 

188.  Waiter,  your  bill  of  fare  amounts  to  so  much,  does  it 
not]  Yes,  sir.  Well,  here  are  so  many  dollars,  and  receipt 
it.  Oh,  sir,  1  off  cannot  be.  Well,  then,  here  is  another  sum 
that  lacks  but  1  dollar  of  the  I  of  the  first;  and  let's  say  no 
more  about  it.  It  is,  nevertheless,  hard,  sir,  to  lose  the  double, 
plus  1,  of  the  square  root  of  my  bill.    What  was  the  amount '^ 

Ans,  144  dollars. 

189.  The  ages  of  the  grandpapa  and  grandson  are  such, 
that  their  quotient  is  equal  to  J  of  their  product,  and  the  sum 
of  the  aforesaid  quotient  and  product  is  320.  What  is  the  age 
of  each  ?  Ans.  96  and  3  years. 

190.  I  mixed  two  pipes  of  wine  ;  one  cost  180  shillings  and 
the  other  140  shillings.  The  first  contained  20  bottles  more 
than  the  second,  and  cost  5d.  less  per  bottle.  What  is  the  value 
of  a  bottle  of  the  mixture?  Ans.  2s.  5d. 

191.  The  quotient  exceeds  the 'divisor  by  half  plus  1,  and 
the  sum  of  the  divisor  and  quotient  exceeds  double  the  square 
root  of  the  dividend  plus  1.  Required  the  dividend,  divisor^ 
and  quotient.  .  Ans.  400,  16,  and  25. 

192.  Two  faucets  running  together,  filled  a  basin  in  3  hours. 
If  the  first  had  run  but  2  hours,  it  would  have  taken  the  second 
6  hours  to  do  the  remainder.  What  time  would  it  take  each 
running  alone  ?  Ans,  4  and  12  hours. 

193.  The  father's  age  has  two  factors,  of  which  one  repre- 
sents  his  daughter's  age,  and  the  other  is  18  less.  Square  this 
other  factor,  add  to  it  ^  of  the  daughter's  age,  and  this  result 
to  the  sum  of  both  ageis,  and  the  general  result  wil^  be  100. 
Required  the  age  of  each.  Ans.  63  and  21. 

194.  The  product  of  two  numbers  exceeds  their  sum  by  14, 
and  their  difference  is  2.     What  are  the  two  numbers  ? 

Ans.  6  and  4. 

195.  A  number  of  three  figures  is  a  multiple  of  11,  and  the 
anits  is  quadruple  the  hundreds.     What  is  the  number? 

Ans.  154. 


APPENDIX.  149 

196.  A  number  of  two  figures  is  such,  that  its  root  placed 
at  the  right  or  left  of  the  aforesaid  number  gives  two  results, 
such,  that  the  latter  exceeds  the  former  by  252.      Ans.  16. 

197.  A  bird-seller,  forgetting  t9  shut  his  coop,  the  greater 
part  of  the  prisoners  availed  themselves  of  the  opportunity, 
and  flew  away.  Being  questioned  about  the  number  of  fugi- 
tives, the  bird-seller  answered ;  If  J^  moje  flew  away,  the  prim- 
itive number,  which , was  over  130  and  under  180,  would 
now  be  reduced  ^.  What  was  the  primitive  number  1  What 
is  their  present  number?  Ans.  156  and  84. 

198.  One  square  is  quadruple  another,  and  their  sum  is  4 
more  than  the  next  square  above  the  greater  of  the  two.  What 
are  the  two  squares'?  Ans.  2p  and  100. 

199.  Lawrence  swops  his  flute  for  Edward's  violin,  and  gives 
to  boot  a  number  of  pounds  equal  to  ^  of  the  price  of  the 
flute  'plus  the  square  root  of  the  price  of  the  violin.  What  is 
the  price  of  each  instrument,  if  the  sum  of  the  two  prices  is 
quadruple  the  pounds  given  to  boot  by  Lawrence  ? 

Ans.  £15  and  £25. 

200.  A  number  of  three  figures  is  such,  that  the  sum  of  its 
digits  is  16;  and  by  inverting  the  number,  then  adding  it  to 
the  number  inverted,  the  sum  will  be  1211  and  the  diflerence 
297.    What  is  the  number  ?  Ans.  457. 

201.  An  oflTicer  gave  the  following  indication  of  the  number 
of  his  regiment :  One  of  its  factors  is  to  the  other  : :  1  :  5,  and 
their  sum  is  to  their  product  : :  6  :  25.  What  was  the  num- 
ber ?  Ans.  125. 

202.  Two  sisters  have  unequal  sums  for  their  purchases. — 
The  elder  has  the  most.  The  sum  of  their  pounds  is  equal  to 
^  of  their  product,  which  latter  would  have  been  ^  less  if  half 
the  pounds  of  the  younger  were  given  to  the  elder.  How  many 
pounds  had  each?  Ans.  £15  and  £10. 

203.  What  is  the  numb.er,  whose  square,  reduced  to  its 
quarter,  excneds  by  |  three  times  the  |-  of  the  number  ? 

Ans.  12. 

204.  Aunt  wants  to  have  a  room  squared,  which  is  2J 
times  as  long  as  it  is  broad.  She  asked  the  mason  how  many 
squares  of  marble  of  a  certain  dimension  would  be  requisite. 
The  mason  answered,  that  if  the  length  was  but  double  the 

13* 


150  APPENDIX. 

breadth,  it  would  have  taken   800  less.     What  do  you  con- 
clude from  this  answer  ? 

Ans.  That  it  would  have  taken  4000. 

205.  A  merchant  gave  ^  of  his  profit  to  the  poor.  At  the 
year's  end  his  alms  amounted  to  $390.  I  demand  what  was 
the  amount  of  his  sales,  if  half  was  at  10,  ^  at  15,  and  the 
remainder  at  18  per  cent,  profit?  Ans.  $60,000. 

206.  If  you  were  sent  to  a  house  whose  number  was  rep- 
resented by  three  figures,  knowing  that  the  digit  representing 
the  hundreds  was  double  that  of  the  tens,  and  that  the  sum 
of  the  three  digits  was  but  ^  of  the  number,  at  what  num- 
ber would  you  rap?  "  Ans.  At  No.  216. 

207.  The  sum  of  the  four  terms  of  a  proposition  is  63 ; 
the  first  is  4  more  than  the  second ;  the  quotient  of  the  third 
by  the  second  is  8^ ;  and  the  product  of  the  means  is  136. 
Required  the  four  terms.  Ans.  8  :  4  :  :  34  :  17. 

208.  The  product  of  the  four  terms  of  a  proportion  is  576 ; 
the  difference  between  the  first  and  the  fourth  term  is  10, 
and  the  quotient  of  the  third  by  the  second  is  f .  What  are 
the  four  terms  ?  Ans.  2  :  6  : :  4  •  12. 


THK  EHD. 


BOOKS  PUBLISHED  BY  D.  4  J.  SADLIER  <fe  COMPANY. 
GREAT  SUCCESS   OF   THE 

The  Volumes  of  the  Library  publvihed  are  the  most  interesting  as 

well  as  the  most  useful  Books  yet^sued  from  the  American 

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Books  which  they  can  safely  place  in  the  hands  of 

their  children. 


(EIGHTH  THOUSAND  JUST  PUBLISHED.) 


T 


he  Popular  liibrary 

OF  HISTORY,  BIOGRAPHY,  FICTIOIN^,  and  MIS- 
CELLANEOUS LITERATURE.  A  Series  of  Works 
by  some  of  the  most  eminent  writers  of  the  day  ;  Edited 
by  Messrs.  Capes,  Northcote,  and  Thompson. 

The  "  Popular  Library  "  is  intended  to  supply  a  desideratum  which 
has  long  been  felt,  by  providing  at  a  cheap  rate  a  series  of  instructive 
and  entertaing  publications,  suited  for  general  use,  wr'tten  expressly 
for  the  purpose,  and  adapted  in  all  respects  to  the  circumstances  of 
the  present  day.  It  is  intended  that  the  style  of  the  wcrk  shall  be 
such  as  to  engage  the  attention  of  young  and  old,  and  of  all  classes 
of  readers,  while  the  subjects  will  be  so  varied  as  to  render  the  series 
equally  acceptable  for  Home  use.  Educational  purposes,  or  railway 
reading. 

The  following  are  some  of  the  subjects  which  it  is  proposed  to  in- 
clude in  the  "  Popular  Library,"  though  the  volumes  will  not  neces- 
sarily be  issued  in  the  order  here  given.  A  large  portion  of  the  Series 
will  also  be  devoted  to  works  of  Fiction  and  Entertaining  Literature 
generally,  which  will  be  interspersed  with  the  more  solid  publications 
here  named. 


F» 


abiola ;  or  the  Church  oFthe  Catacomhsi. 

By  his  Eminence  Cardinal  Wiseman. 

Price. — Cloth,  extra.     12mo.,  400  pages,  $    15 
Gilt  edges,  -  -  -      1  12 

The  Press  of  Europe  and  America  is  unanimous  in  praise  of  this 
work.     We  give  a  few  extracts  below : — 

"  Eminently  popular  and  attractive  in  its  character,  *  Fabiola '  is  in 
many  respects  one  of  the  most  remarkable  works  in  the  whole  range 
of  Modern  Fiction.  The  reader  will  recognise  at  once  those  charac- 
teristics which  have  ever  sufficed  to  identify  one  illustrious  pen." — 
Dublin  Review. 

"  We  rejoice  in  the  publication  of  '  Fabiola,'  as  we  conceive  it  the 
commencement  of  a  new  era  in  Catholic  literature." — Telegraph. 

*'  Worthy  to  etafld  among  the  highest  in  this  kind  of  literature." — 
C.  Standard. 


BOOKS  PUBLISHED  BY  D.  &  J.  SADLIER  <fe  COMPANY. 


"  Were  we  to  speak  of  '  Fabiola '  in  the  strong  terms  our  feelings 
would  prompt,  we  should  be  deemed  extravagant  by  those  who  have 
not  read  it.  It  is  a  most  charming  book,  a  truly  popular  work,  and 
alike  pleasing  to  the  scholars  and  general  reader." — Brownson's  Re- 
view. 

"  A  story  of  the  early  day^f  Christianity,  by  Cardinal  Wiseman, 
is  a  sufficient  notice  to  give  of  this  volume,  lately  published  in  Lon- 
don, and  re-published  by  the  Sadlier's  in  a  very  neat  and  cheap  vol- 
ume."— ^iV.  Y.  Freeman^ s  Journal. 

"  As  a  series  of  beautifully  wrought  and  instructive  tableaux  of 
Christian  virtue  and  Christian  heroism  in  the  early  ages,  it  has  no 
equal  ia  the  English  language." — American  Celt. 

"  We  think  that  all  who  read  *  Fabiola '  will  consider  it  entirely 

•juccessful We  must  do  the  Messrs.  Sadlier  the  justice  to  say, 

that  the  book  is  beautifully  printed  and  illustrated,  and  that  it  is  one 
of  the  cheapest  books  we  have  seen." — Boston  Pilot. 

"  We  would  not  deprive  our  readers  of  the  pleasure  that  is  in  store 
for  them  in  the  perusal  of  *  Fabiola ; '  we  will  therefore  refrain  from 
any  further  extracts  from  this  truly  fascinating  work.  We  know,  in 
fact,  no  book  which  has,  of  late  years,  issued  from  the  press,  so  wor- 
thy of  the  attention  of  the  Catholic  reader  as  '  Fabiola.'  It  is  a  most 
charming  Catholic  story,  most  exquisitely  told." — True  Wit7iess  (Mon 
treal.) 

"  It  is  a  beautiful  production — the  subject  is  as  interesting,  as  is  the 
ability  of  the  author  to  treat  of  it  unquestioned — and  the  tale  itself 
one  of  the  finest  specimens  of  exquisite  tenderness,  lofty  piety,  great 
erudition,  and  vast  and  extended  knowledge  of  the  men  and  mannersi 
of  antiquity,  we  have  ever  read." — Montreal  Transcript. 

'^  As  a  faithful  picture  of  domestic  life  in  the  olden  time  of  Roman 
splendor  and  prosperit}^  it  far  exceeds  the  Last  daj^s  of  Pompeii ;  and 
the  scenes  in  the  arena,  where  the  blood  of  so  many  martyrs  fertilized 
the  soil  wherein  the  seed  of  the  Christian  faith  was  fully  planted,  are 
highly  dramatic,  and  worthy  of  any  author  we  have  ever  read." — 
N'em  York  Citizen. 

*^*  This  volume  is  in  process  of  translation  into  the  French,  Ger- 
man, Italian,  Flemish  and  Dutch  language. 


Ti 


2d  VOL.  POPULAR  LIBRARY,  SER. 

he  liife  of  St.  Francis  of  Rome  ; 

Blessed  Lucy  of  Narni ;  Dominica  of  Baradiso  ;  and  Anne 
of  Montmorency,  Solitary  of  the  Pyrenees.  By  Lady  Ful- 
LERTON.  With  an  Essay  on  the  Miraculous  Life  of  the 
Saints,  by  J.  M.  Capes,  Esq. 

Price. — Cloth,  extra.     12nio.         -  -        50  cents. 

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''  Life  of  St.  Francis  of  Rome,  by  Lady  Georgiana  FuUerton,  together 


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with  some  lesser  sketches,  is  the  second  volume  of  the  series,  and  one 
that  will  be  read  with  delight  by  those  whose  faith  is  untarnished, 
and  whose  imagination  desires  to  be  fed  by  what  is  pure  and  holy. 
Those  whom  the  Apocalypse  describes  "  without,"  "  those  that  serve 
idols,  and  every  one  that  loveth  and  maketh  a  lie,"  or  who  feed  their 
imaginations  on  what  is  of  earth  or  o^hell,  had  better  not  read  the 
book  for  they  will  not  find  it  to  their  taste." — N.  Y.  Freeman's  Journal. 


C3d  VOL.  POPULAR  LIBRARY,  SER. 
atliolic  liCg^ends. 

Containing  the  following  : — The  Legend  of  Blessed  Sadoc 
and  thij  Forty-nine  Martyrs  ;  the  Church  of  St.  Sabina  ; 
The  Vision  of  the  Scholar  ;  The  Legend  of  the  Blessed  Egedius  ; 
Our  Lady  of  Chartres  ;  The  Legend  of  Blessed  Bernard  and  his 
two  Novices  ;  The  Lake  of  the  Apostles ;  The  Child  of  the  Jew  ; 
Our  Lady  of  Galloro  ;  The  Children  of  Justiniani ;  The  Deliv- 
erance of  Antwerp  ;  Our  Lady  of  Good  Counsel ;  The  Three 
Knights  of  St.  John  j  The  Convent  of  St.  Cecily  ;  The  Knight 
of  Champfleury  ;  Qulima,  the  Moorish  Maiden ;  Legend  of  the 
Abbey  of  Ensiedeln  ;  The  Madonna  della  Grotta  at  Naples  ;  The 
Monks  of  Lerins ;  Eusebia  of  Marseilles ;  The  Legend  of  Placi- 
dus ;  The  Sanctuary  of  Our  Lady  of  the  Thorns  ;  The  Miracle  of 
Typasus  ;  The  Demon  Preacher ;  Catharine  of  Rome  ;  The  Legend 
of  the  Hermit  Nicholas ;  The  Martyr  of  Roeux  ;  The  Legend  of 
St.  Csedmon  ;  The  Scholar  of  the  Rosary  ;  The  Legends  of  St. 
Hubert ;  The  Shepherdess  of  Nanterre. 

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H 


4th  VOL.  POPULAR  LIBRARY,  SER. 

eroines  of  Charity. 

With  a  Preface,  by  Aubrey  De  Yere,  Esq.     12mo.  300  pp. 

1.  TPTE  SISTERS  OF  PROVIDENCE,  founders  of  St.  Mary's 

of  the  Woods,  Indiana. 

2.  JEANNE  BIS  COT,  Foundress  of  the  Community  of  St. 

Agness, 

3.  ANNE  DE  MULEN,  Princess  of  Epinoy,  Foundress  of  the 

Religious  of  the  Hospital  at  Beauge. 

4.  LOUISE  DE  MARILLAC,  (Madame  la  Gras,)  Foundress  of 

the  Sisters  of  Charity;  DUCHESS  OF  AIGUILL0I7, 
niece  of  Cardinal  Richelieu ;  MADAME  DE  POLLAIJOKr 
and  MADAME  DE  LAMOIGNON,  her  Companions; 
MADAME  DE  MIRAMION. 


BOOKS  PUBLISHED   BY    D.  &  J.  SADLIER  <fe  COMPANf. 


5.  MES.    ELIZA  A.   SETON,  Foundress  of  the  Sisters  of 

Charity  at  Emmittsburgh. 

6.  JEANNE  JUGAN,  the  Foundress  of  the  Little  Sisters  of 

the  Poor.  ^     ^ 

Price. — Cloth,  «t^tra.     12nio.     60  cents. 
Gilt  edges,     -  -    76     " 

M.  Diipin,  President  of  the  French  Academy,  remarked,  when  award- 
ing  her  the  prize  for  virtue  : — 

**  But  there  remains  a  difficulty,  which  has  doubtless  suggested  itself 
to  the  mind  of  each  amongst  you,  how  is  it  possible  for  Jeanne  alone 
to  provide  the  expenses  of  so  many  poor  ?  What  shall  I  reply  to  you  ? 
God  is  Almighty  I  Jeanne  is  indefatigable,  Jeanne  is  eloquent,  Jeanne 
has  prayers,  Jeanne  has  tears,  Jeanne  has  strength  to  work,  Jeanne 
has  her  basket,  which  she  carries  untiringly  upon  her  arm,  and  which 
she  always  takes  home  filled.  Saintly  woman,  the  Academy  deposits 
in  this  basket  the  utmost  placed  at  its  disposal,  decreeing  you  a  prize 
of  3,000  francs." 

"  This  is  another  of  the  Popular  Library  that  the  Sadliers  are  doing 
such  good  service  by  .re-publishing  in  this  country.  At  the  earnest 
advice  of  a  gifted  friend,  whose  opinion  has  every  weight  with  us, 
we  publish  entire  the  preface  to  this  volume  by  Aubrey  de  Yere.  It 
is  one  of  the  most  interesting  and  valuable  essays  that  has  been  given 
to  the  world  for  a  long  time.  For  the  remaining  contents  of  the  book 
we  must  refer  our  readers  to  its  own  pages." — N.  Y.  Freemari's  Journal. 


^^^  5th  VOL.  POPULAR    uBRARY  SER. 

1  he  IVitch  of  miltoii  Hill. 

A  Tale,  by  the  Author  of  *'  Mount  St.  Lawrence,"  "  Mary, 
Star  of  the  Sea."- 

Price. — Cloth,  extra.     12mo.     60  cents. 
Gilt  edges,      -  -    75     " 

"The  Tales  of  this  Author  are  distinguished  by  a  skill  in  the  deline- 
ation of  character,  and  life-like  pictures  of  domestic  society.  The 
present,  though  bearing  clear  marks  of  the  same  hand,  is  more  astory 
of  movement  and  incident.  It  is  in  no  sense  a  'religious'  novel  and 
still  less  a  controversial  one.  But  the  progress  of  the  story  is  through- 
out made  to  depend  upon  the  practical  adoption  or  reception  of  cer- 
tain principles  of  right  and  wrong,  so  that  the  'poetical  justice' 
awarded  to  every  body  in  the  end  is  not  a  mere  chance  consequence 
of  circumstances,  but  the  natural  result  of  their  character  and  con- 
duct. This  it  is  which  gives  their  chief  value  to  books  of  fiction  as  a 
means  for  influencing  the  reader's  mind,  and  this  it  is  which  will  make 
*The  Witch'  a  volume  fit  for  general  circulation  to  an  extent  which 

•can  be  asked  for  few  novels The  story  is  worked  out  with 

considerable  ingenuity  and  a  rapid  succession  of  events,  increasing  in 

interest  as  it  proceeds '  The  Witch '  is  a  very  clever  story, 

and,  will  suit  the  young  as  well  as  the  old." — Tablet. 


p 


BOOKS  PUBLISHED  BY  D.  <fe  J.  SADLIER  <fe  COMPANY. 

eth  VOL.  POPULAR  LIBRARY  SER. 

ictures  or  Christian  Heroism. 

With  a  Preface  by  the  Rev.  H.  E.  Manning. 

CONTENTS. 


^n  The  Martyrs  of  the  Carmes. 

8.  Gabriel  de  Naillac, 

9.  Margaret  Clitherow, 

10.  Geronimo  at  Algiers, 

11.  Martyrdoms  in  China, 

12.  Father  Thomas  of  Jesus. 


1.  Father  Azeved:>, 

2r  Sister  Honoria  Magaen, 

3*  Blessed  Andrew  Bobola, 

4.  Blessed  John  de  Britto, 

5.  The  Nuns  of  Minsk, 

6.  A  Confessor  of  the  Faith,  1793. 

Price. — Cloth,  extra.     12nio.     60  cents. 
Gilt  edges,     -  -    75     " 

"  The  record  of  the  Martyrdom  of  the  lovely  Father  Azevedo  and 
his  Companions,  who  suffered  most  of  them  in  the  bloom  of  their 
years,  is  one  of  the  most  delightful  pictures  of  Christian  heroism  that 

ever  came  under  our  eyes FeW-  will  read  the  touching  narrative 

of  the  Massacre  of  the  Carmes  without  feeling  their  hearts  melt .... 
"We  have  seldom  seen  so  much  information  within  so  short  a  compass 
on  the  state  of  religious  belief  in  the  Chinese  Empire  as  is  given  in 

the  introductory  chapter  to  the  account  of  the  Chinese  Martyrs 

. . ,  .We  take  leave  of  this  most  interesting  and  edifying  work,  and  we 
recommend  it  most  heartily  to  the  Catholic  reader  who  loves — and 
who  does  not? — to  contemplate  those  pictures  of  heroism  which  the 
Church  alone  has  fed  and  called  forth." — Tablet. 


--^  7th  VOL.  POPULAR  LIBRAPY  SER. 

JtSlakes  and  Flanagans. 

A  Tale  of  the  Times,  Illustrative  of  Irish  Life  in  the  United 
States.  By  Mrs.  J.  Sadlier,  Author  of  "  ^N'ew  Lights,  or 
Life  in  Galway.'^  "  Willy  Burke,"  "Alice  Kiordan,"  &.c. 
12mo.,  400  pages. 

Price.^-Cloth,  extra,      75  cents. 
Gilt  edges,  $1  12     " 

In  a  recent  visit  to  Montreal,  the  Rt.  Rev.  Mgr.  Charbonnell,  preach- 
ing on  the  education  of  children,  paid  the  following  high  compliment 
to  our  Tale — "  The  Blakb6  and  Flanagans" — which  turns  mainly  ou 
that  topic.     His  Lordship  remarked  : — 

"  On  coming  to  town  I  called  at  a  Bookstore  to  purchase  a  number 
of  copies  to  bring  with  me  to  Quebec,  but  found  that  it  was  not  yet 
printed  in  book-form.  It  is  now  being  published  in  the  American 
Celt,  and  I  would  like  to  see  it  circulated  by  the  hundred  thousand." 

"  This  compliment  coming  from  the  Bishop  of  Toronto,  cannot  be  but 
highly  valued  by  the  gifted  writer,  Mrs.  Sadlier. 

*'  We  cannot  forbear  assuring  the  Authoress,  contemporaneous  with 


BOOKS  PUBLISHED  BY  D.  &  J.  SADLIER  &  COMPANY. 


announcement,  and  thus  publicly,  that  her  story  has  been  read  with 
the  liveliest  interest  by  our  readers,  and  that  it  has  elicited  the  most 
earnest  expressions  of  respect  for  herself  from  persons  distinguished 
in  every  walk  of  life. 

"Before  parting  with  our  favorite  contributor,  let  us  add  the  fol- 
lowing opinion  recently  expresf'^d  to  us  by  a  gifted  Western  clergy- 
man 1  *  Mrs.  Sadlier's  story,'  he  said,  *  has  done  more  to  bring  hoine 
to  the  hearts  of  parents  the  importance  of  Catholic  education,  than 
any  other,  and  all  other  advocacy  combined.'  This  is  extreme  praise, 
but  we  will  say,  not  undeserved." — American  Celt. 

"  The  style  is  excellent,  thoroughly  natural  and  unaffected,  the  nar- 
rative flowing,  the  conversations  full  of  vivacity,  and  the  characters 

well  sustained We  cannot  but  wish  it  the  widest  circulation 

that  a  book  can  have." — >S'^.  Zouis  Leader, 


p— ^  8th  VOL.  POPULAR  LIBRARY  SER. 

X  he  liife  and  Tiiiiei§i  of  St.  Bernard. 

Translated  from  the  French  of  M.  L'Abbe  IIatisbonne. 
With  a  Preface  by  Henry  Edward  Manning,  D.  D.,  and 
a  Portrait.     1  vol.,  12mo.,  500  pages. 

Price. — Cloth,  extra,  $1  00. 
Gilt  edges,       1  50. 

"  St.  Bernard  was  so  eminently  the  saint  of  his  age,  that  it  would 
be  impossible  to  write  his  life  without  surrounding  it  with  an  exten- 
sive history  of  the  period  in  which  he  lived,  and  over  which  he  may 
be  truly  said,  to  have  ruled.  The  Abbe  Ratisbonne  has,  with  this 
view, 'very  ably  and  judiciously  interwoven  with  the  personal  narra- 
tive and  description  of  the  ^aint,  the  chief  contemporaneous  events 
and  characters  of  the  time. 

"  There  seems  to  have  been  in  this  one  mind  an  inexhaustible  abund- 
ance, variety,  and  versatility  of  gifts.  Without  ever  ceasing  to  be 
the  holy  and  mortified  religious,  St.  Bernard  appears  to  be  the  ruling 
will  of  his  time.  He  stands  forth  as  pastor,  pr cache"",  mystical  writer, 
controversialist,  reformer,  pacificator,  mediator,  arbiter,  diplomatist, 
and  statesman.  He  appears  in  the  schools,  at  the  altar,  in  the  preach- 
er's chair,  in  councils  of  the  Church,  in  councils  of  tlie  State,  amid 
the  factions  of  cities,  the  negotiations  of  princes,  and  the  contests  of 
anti -popes.  And  whence  came  this  wondrous  power  of  dealing  with 
affairs  and  with  men  ?  Not  from  the  training  and  schooling  of  this 
world,  but  from  the  instincts,  simplicity,  and  penetration  of  a  mind 
profoundly  immersed  in  God,  and  from  a  will  of  which  the  fervour 
and  singleness  of  aim  were  supernatural." — Extracts  from  Preface. 


BOOKS  PUBLISHED  BY  D.  &  J.  SADLIER  «fe  COMPANY. 


-^^  9th  VOL.  POPULAR  LIBRARY  SEP. 

J.  he  ffiiie  and  Victories  of  the  Karly  ilJar- 

TYRS.  By  Mrs.  Hope.  Written  for  the  Oratorian's 
Schools  of  our  Lad3^'s  Compassion.  1  vol.,  12mo.,  400 
pages.  •« 

Price. — Cloth,  extra.   $15. 
Gilt  edges,         1  12. 

"  Tlie  interesting  Tale  of  *  Fabiola '  has  made  most  readers  familiar 
with  the  sufferings  of  the  Early  Martyrs,  and  desirous  to  know  more 
of  their  history,  and  of  the  Victories  which  they  achieved  over  the 
world.  Every  age,  every  clime,  has  its  martyrs,  for  it  is  a  distinctive 
mark  of  the  Catholic  Church,  that  the  race  of  martyrs  never  dies  out. 
And  since  her  earliest  times,  a  single  generation  has  not  passed  away 
without  some  of  her  children  shedding  their  blood  for  the  name  of 
Jesus.  Other  religious  bodies  may  have  had  a  few  individuals  here 
and  there  and  at  distant  intervals,  who  have  died  for  their  opinions. 
But  it  is  in  the  Catholic  Church  alone  the  spirit  of  martyrdom  has 
ever  been  alive.  Nor  is  it  difficult  to  account  for  this,  as  the  Catholic 
Church  is  the  only  true  Church,  the  devil  is  ever  ready  to  raise  up 
persecutions  against  her,  and  as  the  Lord  ever  loves  his  spouse,  the 
Church,  he  bestows  upon  her  all  graces,  and  among  them  the  grace  of 
martyrdom,  with  a  more  lavish  hand  than  on  others." — Extract  from 
Jo\trodaction. 

10th  VOL.  POPULAR  LIBRARY,  SER. 

History  of  the  War  in  lia  Vendee,   and 

the  Little  Chouannerie.     By  G.  J.   Hill,  M.  A.     With 

Two  Maps  and  Seven  Engravings,  12mo. 

Cloth,  extra,  75  cents;  cloth,  extra,  gilt  edges,  $1  12. 

Til  is  is  a  number  of  the  popular  Catholic  Library  now  in  course  oP 
publication  by  Messrs.  Burns  and  Lambert,  London,  and  Messrs. 
Sadlier  <fe  Co.,  in  this  city,  and  is  one  of  the  most  interesting  num- 
bers of  that  valuable  series  that  has  as  yet  a]ipeared.  It  is  full  of 
romantic  incident,  and  is  as  exciting  as  any  romance  ever  written. 
The  author  has  grouped  his  incidents  with  much  skill,  and  told  his 
story  with  much  grace  and  feeling.  The  public  can  hardly  fail  to  ap- 
preciate it  highly ;  and  the  intelligent  reader  will  find  it  full  of  in- 


BOOKS    PUBLISHED    BY    D.    &    J.    SADLIER    &    COMPANY. 

struction  as  well  as  interest.  The  war  in  La  Vendee  was  the  war  of 
the  peasants,  and  a  war  chiefly  for  the  freedom  of  religion.  For  the 
freedom  of  religion,  the  freedom  of  conscience,  the  freedom  to  have 
their  own  priests,  and  to  worship  God  according  to  their  own  faith, 
they  took  up  arms,  and  they  di€rnot  lay  them  down  till  they  had  se- 
cured it.  They  were  the  true  French  patriots  in  the  time  of  the  Re- 
public ;  the  men  who  preserved  fresh  and  living  the  France  of  St. 
Louis  and  the  Crusaders,  historical  and  traditional  France,  and  it  is 
not  too  much  to  say  that  they  saved  France  in  her  darkest  da3"s,  and 
prevented  the  continuity  of  her  life  from  being  broken  by  the  madness 
and  excesses  of  the  revolution.  They  prove  how  much  religion  warms 
and  strengthens  patriotism,  and  that  it  can  make  undisciplined  peas- 
ants able  to  cope  successfully  with  the  best  drilled  and  appointed 
armies.  When  fighting  for  their  religion,  these  half-unarmed  peasants 
were  invincible,  and  almost  always  victorious.  When  from  a  Catholic 
army  they  became  a  Royalist  army,  and  acted  under  the  direction  of 
chiefs,  who  thought  only  of  restoring  fallen  monarchy,  they  melted 
away  before  their  enemies,  as  wax  at  the  touch  of  fire,  or  were  scat- 
tered as  the  morning  mist  before  the  rising  sun. — BrownsorCs  Review. 

The  period  of  the  war  was  from  *91  to  '97,  from  the  first  proclama- 
tion of  "the  Civil  Constitution  of  the  Clergy"  to  the  pacification,  and 
consequent  restoration  of  public  worship  under  the  Consulate.  The 
scene  of  the  war  was  the  most  Celtic  region  in  the  west  of  France,  on 
both  banks  of  the  great  Celtic  river,  the  Loire.  All  Bretagne,  parts 
of  Maine,  Anjou  and  Poitou,  were  engaged  at  various  p-eriods  of  the 
struggle,  the  lai*gest  number  in  arms  being  not  less  than  100,000  in 
every  quarter  of  this  country,  while  against  them  were  sent  in  succes- 
sion the  armies  of  Generals  Kleber,  Westerman,  and  Hoche.  On  the 
side  of  the  insurgenjts,  the  successive  commanders  were  Cathelineau, 
D'Elbee,  Henri  de  Larochejacquelein,  Charette,  and  Stofilet;  St.  Malo, 
^iuiberon,  JSTantes,  Laval,  Moulins,  Fontenay,  Thouars,  and  Angers, 
were  the  towns  and  cities  lost  and  taken  on  both  sides,  but  the  citadel 
and  nursery  of  the  war  was  La  Vendee. — ^American  Celt. 


11th  VOL.  POPULAR  LIBEAEY,  SEE. 

X  ales  and  liegends  iroin  Hiistory. 

Contents. — 1.  Gonsalvo  of  Amaranta.  2  The  Victory 
of  JMiiret.  3.  The  Dominicans  in  Ghent.  4.  The  Martyrs 
of  Stone.  5.  The  Abbey  of  Premontre.  6.  Legenis  of 
St.  Winifrile.     7.  The  Feast  of  the  Immaculate  Ccmcep- 


BOOKS    PUBLISHED    BY    D.    <b    J.    SADLIER    A    COMPANY. 

tion.  8.  The  Consecration  of  Westminster  Abbey.  9. 
The  Monk's  Last  Words.  10.  The  Martyr  Maidens  of 
Ostend.  11.  The  Loss  of  the  '*  Conception.'^  12.  Foun- 
dation of  the  Abbey  of  Anchin.  13.  Our  Lady  of  Mercy. 
14.  John  de  la  Cambe.  15.  Th? Carpenter  of  Roosendael. 
16.  The  Widow  of  Artois.  17.  The  Yillage  of  Blanken- 
berg.  18.  St,  Edward's  death.  19.  The  Windows  of 
San  Petronio.     20.  The  Vessels  of  St.  Peter. 

12mo.,  cloth,  extra,  63  cents;  cloth  gilt,  88  cents. 


12th  VOL.  POPULAK  LIBPwAEY,  SEE. 

The  Missions  in  Japan  and  Paraguay. 

By  Cecilia  Caddell,  author  of  "  Tales  of  the  Festivals/' 
"  Miser's  Daughter,"  "  Lost  Jenoveffa,"  &c.,  &c. 

12mo.,  cloth,  extra,  63  cts. ;  cloth,  extra,  gilt  edges,  88  cts. 
These  are  two  additional  numbers  of  the  Catholic  Library  already 
referred  to,  and  are  well  adapted  to  its  purpose.  The  History  of  the 
Missions  in  Japan  and  Paraguay,  by  Miss  Caddell,  is  peculiarly  inter-' 
esting  and  instructive.  It  is  written  with  simplicity  and  taste. — 
Brownson^s  Revieio. 


13th  VOL.  POPULAR  LIBRAEY,  SEE. 

Calllsfa;  A  TaJe  of  the  Third  Century. 

By  Yery  Rev.  John  Henry  Newman,  D.  D.,  Rector  of 

the  Catholic  University,  Dublin. 

12mo.,  cloth,  extra,  75  cents;  cloth,  extra  gilt,  $1  12. 

The  following  extract  is  from  a  very  long  notice  of  the  work  in  the 
Dublin  Tablet: 

*'  The  story  is  partly  interwoven  with  historical  facts,  but  its  author 
professes,  at  the  outset,  that  as  a  whole,  it  is  'a  simple  fiction  from 
beginning  to  end.'  However  that  may  be,  as  an  instrument  of  convey- 
ing a  real  and  genuine  historical  knowledge  of  the  days  of  which  it 
treats,  in  their  aspect  towards  Christianity,  it  will  probably  remain 
without  a  rival  in  the  literary  world." 

Dr.  Newman  is  peculiarly  felicitous  in  portraying  the  exceeding  dif- 
ficulty an  old  Roman  Pagan  had  in  understanding  the  principles  and 
motives  w^hich  governed  the  Christian.  He  could  not  enter  into  his 
way  of  thinking,  could  not  seize  his  stand-point,  and  comprehend  ht)W 
a  man  well  to  do  in  the  world,  intelligent,  cultivated,  respectable,  in 


BOOKS    PUBLISHED    BY    D.    &l    J.    SADLIER    <fe    COMP^Nr. 

the  road  to  wealth,  pleasure,  distinction,  honors,  under  Paganism, 
shonld  forego  all  his  advantages,  join  himself  to  a  proscribed  sect,  re- 
garded as  almost  beneath  contempt,  professing,  as  it  was  assumed,  a 
mo^it  debasing  and  disgusting  miperstition,  without  intelligence,  com- 
mon sense,  and  common  decency,  and  that  too  when  to  do  so  was  sure 
persecution,  and  almost  certain  death,  as  a  traitor  to  Caesar.  It  was 
a  sore  puzzle  to  the  wise  heathen,  it  is  a  sore  puzzle  also,  to  the  proud 
an4  self-sufficient  non-Catholic  even  to-day,  and  will  remain  so  to  the 
end  of  the  world.  It  is  hard  even  for  worldly-minded  Christians  to 
comprehend  how  youth  and  beauty  can  forego  the  world,  and  wed 
themselves  to  an  unseen  Lover,  and  live  and  suffer  only  for  an  in- 
visible Love.  The  solution  is  found  only  in  giving  our  hearts  to  God, 
and'living  for  heaven  alone." — BrownsorCs  Heview." 


14th  VOL.  POPULAR  LIBEARY,  SEE. 

Manual  of  Modern  Hii^tory. 

Bj  Matthew  Bridges,  Esq.     600  pages. 

12mo.,  cloth,  or  half  roan,  $1  00. 

This  volume,  containing,  as  it  does,  a  large  amount  of  matter,  with 
complete  indexes,  tables  of  chronology,  <fec.,  &c.,  will  be  found  equally 
useful  for  popular  reading,  as  a  Student's  Text-book,  or  as  a  manual 
for  schools. 

"Mr.  Bridges' excellent  Popular  Modern  History." — Card'mal  Wise- 
man^ s  late  Lecture  on  Ro7ne. 

"The  author  dates  this  history  from  the  irruption  and  settlement  of 
the  barbarous  nations  on  the  ruins  of  the  Western  Empire,  and  brings 
it  down,  noting  events  as  they  transpired  in  the  succeeding  ages,  to 
the  present.  The  author  has  carefully  compiled  the  work  from  such 
resources  as  were  within  his  reach,  and  he  considered  were  of  a  re- 
liable character." — Pittsburgh  Catholic.^^ 


15th  VOL.  POPULAR  LIBRARY,  SEB. 

Ancient  History. 

By  Matthew  Bridges,  Esq. 

Price,  ^5  cents. 

This  volume,  containing,  as  it  does,  a  large  amooit  of  matter,  with 
complete  indexes,  tables  of  chronology,  <fcc.,  <fec.,  will  be  found  equally 
useful  for  popular  reading,  as  a  Student's  Text-bock,  or  as  a  manual 
for  schools. 


BOOKS   PUBLISHED    BY    D.    <fe   J.    SADLIER    «fe  COMPANY. 


16th  VOL.  POPULAR  LIBRARY  SER. 

Life  an»  i.abor8  of  st.  vincewt 

©E  PAUIi.     A  new,  complete,  and  careful  Biography. 

By  H.  Bedford,  Esq.     12mo.  ^ 

Clotli^  extra 50  cU. 

Cloth^  extra  gilt 75*  " 

"  The  loving  warmth  which  pervades  this  book — the  accuracy 
of  detail  with  which  it  is  written — and,  above  all,  the  subject — 
one  of  God's  greatest  saints  and  the  world's  greatest  benefactors — 
must  ensure  it  a  wide  circulation  in  America.  For  the  rest,  the 
publishers  have  got  it  out  in  good  style,  on  good  paper,  with  clear 
type. " — Rami  ler. 

17th  VOL.  POPULAR  LIBRARY  SER. 


Ai 


XICE  SHERWI?¥. 

An  Historical  Tale  of  the  Days  of  Sir  Thomas  More. 

12mo. 

Gloth^  extra $0  75 

Glot\  extra  gilt 1  12 

"  Alice  Sherwin". — This  able  and  interesting  historical  novel  is 
reprinted  from  the  English,  and  has  been  ascribed,  we  know  not 
whether  justly  or  not,  to  the  distinguished  author  of  Sunday  in 
London — a  convert  from  Anglicanism,  who  deserves  the  thanks 
of  every  English  speaking  Catholic  for  the  valuable  contributions 
he  has  made  since  his  conversion,  and  is  still  making,  to  English 
Catholic  literature.  But  by  whomsoever  written,  Alice  Sherwin 
is,  so  far  as  we  know,  the  most  successful  attempt  at  the  genuine 
historical  novel  by  a  Catholic  author  yet  made  in  our  language, 
and  gives  goodly  premise  that,  in  due  time,  we  shall  take  our 
proper  rank  in  this  department  of  literature,  rendered  so  popular 
by  the  historical  romances  of  Sir  Walter  Scott.  The  author  has  a 
cultivated  mind,  a  generous  and  loving  spirit,  and  more  than  usual 
knowledge  of  the  play  of  the  passions  and  the  workings  of  the 
human  heart.  He  has  studied  with  care  and  discernment  the 
epoch  of  Sir  Thomas  More,  or,  as  we  prefer  to  say,  of  Henry  YI|I., 
and  has  successfully  seized  its  principal  features,  its  costume  and 
manners,  and  its  general  spirit,  and  paints  them  in  vivid  colors, 
and  with  a  bold  and  free  pencil,  though  after,  all  with  more  talent 
and  skill  than  genius  in  its  highest  sense." — Brownson's  Review, 


BOOKS   PUBLISHED   BY   D.    &  J.    SADLIER   &  OOMPAlTr. 


18th  VOL.  POPtTLAR  LIBRARY  SERIES. 

Xhe    liife    of  St.    Franciis   de    Sales, 

Bishop  and  Prince  of  Geneva.     By  Robert  Ornsby, 
M.  A.   '  Price,  cloth  iKtra,  63c ;  gilt,  88c. 
From  tlie  Pr^ace  of  this  excellent  worTc  we  make  tlie  following  extract. 
**  The  Saint  whose  life  is  treated  of  in  this  volume  is  perhaps— next,  of 
course,  to  the  Queen  of  Saints — the  '■  favorite  Saint'  of  the  whole  calendar, 
wherever  his  writings  are  known  and  understood.     There  appears  in  the 
mind  of  St.  Francis  of  Sales  that  union  of  sweetness  and  strength  of  manly 
power  and  feminine  delicacy,  of  profound  knowledge  and  practical  dexter- 
ity, which  constitute  a  character  formed  at  once  to  win  and  subdue  minds 
of  almost  every  type  and  age.     As  the  rose  among  flowers,  so  is  he  among 
Saints.  From  the  thorny,  woody  fibre  of  the  brier  comes  forth  that  blossom 
which  unites  all  that  can  make  a  flower  lovely  and  attractive ;  and  from  the 
hot  and  vehement  nature  of  the  young  Savoyard,  came  a  spiritual  bloom 
whose  beauty  and  fragrance  were  perfect  in  an  extraordinary  degree." 

"  When  we  say  that  the  author  of  the  work  before  us  has  done  this,  it  is  to 
say  that  he  has  done  all  that  a  human  pen  can  do  in  portraying  such  a 
saintly  figure.  Human  effort  must  ever  be  as  far  below  the  lustrous  reality 
the  gold  with  which  our  painters  halo  the  sacred  heads  that  adorn  our 
churches  is  below  the  brightness  of  the  divine  rays.  Mr.  Ornsby  has  done 
what  a  writer  can  do.  The  pious  imagination  of  his  Catholic  readers  will 
aid  in  giving  life  to  his  portrait.  Selections  from  the  'Spirit  of  St.  Francis 
de  Sales,'  by  the  Bishop  of  Belley,  are  added  to  the.  biography.  Recom- 
mendation of  tl^em  to  the  perusal  of  our  readers  and  their  friends  is  need- 
less.'— New  York  Tablet* 


19th  VOL.  POPULAR  LIBRARY  SERIES. 

ihe  liife  of  St.  Elizabeth  of  Mun^ary^ 

Duchess  of  Thuringia.  By  the  Count  de  Montalem- 
BERT,  peer  of  France.  Translated  from  the  French, 
hy  Mary  Hackett  and  Mrs.  J.  Sadlier.  1  Vol. 
Eoyal  12mo.  Fine  paper,  with  a  splendid  Portrait  after 
Overbeck,  engraved  on  steel.  Price,  Cloth  extra,  $1 
and  $1.50;  English  moi^.,  extra,  $2. 

In  the  translation  of  this  work,  published  in  Dublin,  the  introduction, 
which  forms  about  150  pages  of  the  original,  and  nearly  one-third  of  the 
whole  work,  was  omitted.    Our  edition  contains  the  entire  work. 

This  is  one  of  the  most  interesting  biographies  in  any  language.  We 
defy  even  the  most  lukewarm  Catholic  to  read  it  without  inwardly  thanking 
God  that  he  belongs  to  a  Church  that  can  produce  such  purity  and  holiness, 
and  so  much  humility  and  self-denial  as  is  exemplified  in  the  life  of  thf 
"  dear  St.  Elizabeth." 

Postage,  24  CentB. 


BOOKS  PUBIJSHED  BY  D.  !f  J.  SADLIER  k  COMPANY. 

Sablitr  s  Jfiits&e  f  ilirarg. 

No.  1.    The  Orphan  of  Moscow. 
OR,  THE  YOUNG  GOYERMESS.    A  Tale.    Translated 

from  the  French  of  Madame  Woillez,  by  Mrs.  J.  Sadlier,  Illustrated 
with  a  Steel  Engraving  and  an  Illuminated  Title.  Full  cloth,  gilt 
backs.  50  cents;  Full  gilt,  gilt  edges,  75  cents;  Imitation  morocco, 
gilt,  $1  00.  P.  12  cts. 

No.  2.    The  Castle  of  Roussillon. 
OR,  QUERCY  IN  THE  SIXTEENTH  CENTURY.    A 

Tale.  Translated  from  the  French  of  Madame  Eugenie  de  la  Rochere, 
by  Mrs.  J.  Sadlier.  Illustrated  with  a  Steel  Engraving  and  an  Illu- 
minated Title.  Cloth,  gilt  back,  50  cents;  Cloth,  gilt  back  and 
edges,  75  cents;  Imitation  morocco,  $1  00.  P.  12  cts. 

P.  8  cts.  No.  3.     Sick  Calls. 

PROM  THE  DAIRY  OF  A  MISSIONARY  PRIEST. 

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The  Destitute  Poor ;  Tlie  Merchant's  Clerk  ;  Death -beds  of  the  Poor; 
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No.  5.    Nevr  Lights:  or,  Life  in  Gal-way. 
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No.  7.    Tales  of  the  Five  Senses. 
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tions in  each,  printed  on  the  finest  paper,  16mo.  vols,, 
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FIRST  SEPJES. 

The  Boyhood  of  Oreat  Painters, 

From  the  French. 

16mo.,  cloth,  extra,  38  cents;  gilt,  63  cents. 
Contents. — Michael  Angelo  Buonarotti — Corregio — Bar- 
tholomew   Esteban    MmMllo — Sebastian     Gomez — David 
Teniers — Anthony      Watteau — Giotto — The      Murillo — 
Painting. 

SECOND  SEEIES. 

X  he  Boyhood  of  Great  Painters. 

From  the  French. 

16mo.,  cloth,  extra,  38  cents;  gilt,  63  cents. 

Contents. — Salvator     Kosa — Perugino — Ribera — Dirk — 

Mozart — Leonardo  Da  Yinci — Rafaelle  D'Urbino — Jac- 
•  qnes  Callot. 

"  We  can  scarcely  give  too  much  praise  to  the  two  handsome 
volumes  before  us,  bearing  this  title  ;  whether  we  consider  the  inter- 
est of  tlie  sketches,  the  style  of  their  composition,  or  the  very  credit- 
able manner  in  which  the  mechanical  portion  of  the  books  has  been 
executed.  We  have  here,  as  it  were,  the  interior  life  of  the  grent 
masters  of  painting ;  the  wayward,  wilful,  indomitable  Salvator  Rosa ; 
the  stern  and  proud  Angelo;  the  vain  and  beautiful  RaffaeU  the 
modest  and  sensitive  Corregio;  the  laborious  Da  Yinci;  the  astute 
Murillo,  and  others,  whose  names  form  a  bright  halo  round  the  brow 
of  the  mother  of  all  that  is  beautiful  in  Art — the  Catholic  Church.'*— 
American  Celt. 

Ihe  miner's  Daug^hter. 

A  Catholic  Tale,  by  Miss  Cecilia  Mary  Caddell. 
16mo.,  cloth,  extra,  38  cents;  gilt,  63  cents. 

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AND  PARABLES  FOR  LITTLE  CHILDREN.    By 

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Instruction  and  Amusement  of  youth.  Translated  from 
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Dublin  Tablet. 

Lost  Oenovelfa  5   or,  the  Spouse  of  the 

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Cadell,  Authoress  of  the  "  Miner's  Daughter,"  &c ,  &c. 

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Little  Joseph  5  or,  the  IToung^  Savoyard, 

AND  OTHER  TALES. 

Contents  — Little  Joseph  ;  or  the  Young  Savoyard.  The 
Orphan  of  Gaeta.  Three  Yisits  to  the  Hotel  des  Invalides 
— chapter  1,  Le  Grande  Monarque»— 2,  a  Hundred  Years 
After — 3,  Sic  Transit  Gloria  Mundi.  The  Singer  of  Ban- 
geres.  Baldwin  the  Ninth.  The  Stuffed  Tigers.  The 
Lamb. 

16mo.,  cloth,  extra,        .        .     $0  38 
gilt,    .        .       0  63 

"These  six  fine  little  volumes  belong  to  Messrs.  Sadlier's  Young 
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ton*  8  Review. 


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-wwy-  NOW   PUBLISHED-THE   COMPLETE 

V?  orks  aiKl  Life  of  Gerald  Griffin,      * 

New  Edition  of  the  LIFE  AND  WORKS  OF  GER- 
ALD GRIFFIN,  revi^d  and  corrected  by  his  Brother. 
Ilkistrated  with  splendid  Steel  Engravings,  and  printed  on 
the  finest  paper.  Complete  in  thirty- three  weekly  parts, 
at  Twenty -five  Cents  each.  Comprising  the  following 
Tales  :— 

VOL.     1.— THE  COLLEGIANS.     A  Tale  of  Garryowen. 
2.— CARD  DRAWING.     A  Tale  of  Clare. 

THE  HALF  SIR.     A  Tale  of  Munster.  ^-^ 

SUIL  DHUV.     A  Tale  of  Tipperary. 
S.^THE  RIVALS.     A  Tale  of  Wicklow ;   and  TRACY'S 

AMBITION. 
4— HOLLAND  TIDE,^  THE  AYLMERS  OF  BALLYAYL- 
MER,  THE  HAND  AND  WORD,  and  BARBER  OF 
BANTRY. 
5.— TALES  OF  THE  JURY  ROOM.     Containing:  SIGIS- 
MUND,   THE  STORY-TELLER  AT  FAULT,  THE 
KNIGHT  WITHOUT  REPROACH,  &g.,  &c. 
6.— THE  DUKE  OF  MONMOUTH.     A  Tale  of  the  English 

Insurrection. 
7.— THE  POETICAL  WORKS  AND  TRAGEDY  OF  GYS- 
SIPUS. 
"         8.— INVASION.     A  Tale  of  the  Conquest. 

9.— LIFE  OF  GERALD  GRIFFIN.     By  his  Brother. 
''       10.— TALES  OF  THE  FIVE  SENSES,  and  NIGHT  AT  SEA. 
^^"  The  Works  will  also  be  bound  in  eloth  extra,  and  issued  in  ten 
monthly  Volumes,  at  ONE  DOLLAR  per  volume.     Sent  free  by  post 
to  any  part  of  the  United  States. 


In  presenting  to  the  American  public,  a  first  edition  of  the  Works 
of  Gerald  Griffin,  the  Publishers  may  remark  that  it  will  be  found  to 
be  the  only  complete  on£.  Neither  in  the  London  nor  Dublin  editions, 
could  the  Publishers  include  the  historical  novel  of  *'  The  Invasion," 
and  the  celebrated  tragedy  of  '*  Gj'ssipus."  As  Ave  are  not  subject  to 
any  restriction  arising  from  the  British  copyright,  we  have  included 
the  former  with  the  prose,  and  the  latter  with  the  poetical  works  of 
the  Author. 

We  are  also  indebted  to  near  relatives  of  Mr.  Griffin,  residing  in  this 
country,  for  an  original  contribution  to  this  edition;  which  will  be 
found  gratefully  acknowledged  in  the  proper  place. 

As  the  life  of  the  Author  forms  the  subject  of  one  entire  volume,' 
we  need  say  little  here,  of  the  uncommon  interest  his  name  continues 
to  excite.      Unlike  the  majority  of  writers  of  fiction,  his  reputation 
has  widely  expanded  since  his  death.     In  1840,  when  he  was  laid  in 
his  grave,  at  the  early  age  of  seven  and  th'irty,  not  one  person  knew 


BOOKS  PUBLISHED  BY  D.  &  J.  SADLTER  &  COMPANY. 


the  loss  a  pure  lii-erature  had  sustained,  for  fifty  who  now  join  vener- 
ation for  his  virtues,  to  admiration  for  his  various  and  delightful  tal- 
ents. The  goodness  of  his  heart,  the  purity  of  his  life,  the  combined 
humor  and  pathos  of  his  writings,  all  promise  longevity  of  reputation 
to  Gerald  Griffin.  O 

"  He  had  kept 
The  whiten/»ss  of  his  soul,  and  so  men  o'er  him  wept." 

He  united  all  the  simplicity  and  cordiality  of  Oliver  Goldsmith  to 
OQueli  of  the  fiery  energy  and  manly  zeal  of  Robert  Burns.  His  life 
docs  not  disappoint  the  reader,  who  turns  from  the  works  to  their 
author ;  it  is,  indeed,  the  most  delightful  and  harmonious  of  all  his 
works.  From  his  childish  sports  and  stories  by  the  JShannon  until  his 
solemn  and  enviable  death  beside  "  the  pleasant  waters  "  of  the  Lee, 
a  golden  thread  of  rectitude  runs  through  all  his  actions.  A  literary 
adventurer  in  London  at  nineteen,  with  a  Spanish  tragedy  for  his  sole 
capital,  famous  at  thirty,  a  religious  five  years  later,  a  tenant  of  the 
Christian  Brothers'  Cemetery  at  thirty-seven — the  main  story  of  his 
life  is  soon  told.  Over  its  details,  we  are  confident,  many  a  reader 
will  fondly  linger,  and  often  return  to  contemplate  so  strange  and  so 
beautiful  a  picture.  Out  of  his  secret  heart  they  will  find  sentiments 
issuing  not  unworthy  of  St.  Frances  de  Sales,  while  from  his  brain 
have  sprung  creations  of  character  which  might  have  been  proudly 
fathered  by  Walter  Scott. 

Canvassers^ wanted  in  every  part  of  the  United  States  and  Canadas 
to  sell  this  work. 

The  Works  are  bound  in  the  following  styles  of  Binding: — 

10  Vols,  cloth  extra,  (when  sold  separate,  only  the  cloth,)  $10  00 

«      "         "         " full  gilt,    15  00 

"      "      Sheep,  Library  style,  or  half  Roan,  cloth  sides,-       12  50 
"      "      Half  morocco,  marble  edges,     -         -         -        -       15  00 

"      "        "     calf,  antique, 20  00 

"      "         "         "         "         or  morocco,  gilt,  -         -       80  00 

NOTICES  OF  THE  PRESS. 

The  Complete  Works  of  Gerald  Griffin. — We  welcome  this  new 
and  complete  edition  of  the  works  of  Gerald  Griffin,  now  in  the  course 
of  publication  by  the  Messrs.  Sadlier  &  Co.^  We  read  The  Collegians, 
when  it  was  first  published,  with  a  pleasure  we  have  never  forgotten, 
and  which  we  have  found  increased  at  every  repeated  perusal.  Ire- 
land haS'  produced  many  geniuses,  but  rarely  one.  upon  the  whole, 
superior  to  Gerald  Griffin.  When  we  have  his  life,  and  the  publica- 
tion of  the  edition  is  completed,  we  shall  endeavor  to  render  our  tri- 
bute of  gratitude  to  the  memory  of  the  gifted  author. — [Browjison's 
Review. 

It  is  not  surprising  that  comparatively  little  is  known  in  this  coun- 
try, save  among  the  more  intelligent  American  citizens  from  the 
''Emerald  Isle,"  of  Gerald  Griff'n,   when  we  consider  that  he  was 


BOOKS  PUBLISHED  BY  D.  &  J.  SADLIER  &  COMPANY. 

born  in^the  beginning  of  the  present  centur\',  and  closed  his  brief, 
but  reall}'  bright  and  brilliant,  career  of  authorship  at  the  early  age 
of  37.  He  however  achieved  a  reputation  as  a  writer  of  no  ordinary 
power,  and  as  has  been  remarkffi,  united  all  the  simplicity  and  cord- 
iality of  Oliver  Goldsmith,  to  much  of  the  fiery  energy  and  manly 
zeal  of  Robert  Burns.  We  have  now  before  us  four  volumes,  the 
commencement  of  a  complete  edition  of  his  works,  embracing  the 
*'  Collegians,^^  and  the  fii^et  series  of  his  "  MunUer  Tales."  The  na-^ 
tionality  of  these  tales,  and  the  genius  of  the  author  in  depicting  the 
mingled  levity  and  pathos  of  Irish  character,  have  rendered  t'frem  ex- 
ceedingly popular.  The  present  edition,  the  first  published  in  Amer- 
ica. The  style  in  which  the  series  is  produced,  is  highly  creditable 
to  the  enterprise  of  the  American  publishers,  and  we  are  free  to  say, 
that  the  volumes  are  worthy  of  being  placed  in  our  libraries,  public 
or  private,  alongside  of  Irving,  Cooper,  or  Scott. — Hunt's  Merchant^ 
Magazine. 

"Whoever  wishes  to  read  one' of  the  most  passionate  and  pathetic 
novels  in  English  literature  Avill  take  with  him,  during  the  summer 
vacation,  The  Collegians,  by  Gerald  Grifhn.  He  was  a  young  Irish- 
man, who  died  several  years  since,  after  writing  a  series  of  works — 
novels  and  poetry,  which  gave  him  little  reputation  during  his  life, 
but  since  his  death  have  given  him  fame,  as,  in  our  judgment,  the  best 
Irish  novelist.  The  .picture  of  Irish  character  and  manners  a  half 
century  since,  in  The  CollegianSy  is  masterly,  and  the  power  with 
which  the  fond,  impetuous,  passionate,  thoroughly  Celtic  nature  of 
Hardress  Cregan  is  drawn,  evinces  rate  genius.  Griffin  died  young, 
a  disappointed  man.  But  this  is  one  story,  if  nothing  else  of  his,  will 
surely  live  among  the  very  best  novels  of  the  time.  It  is  full  of  in- 
cident, and  an  absorbing  interest  allures  the  reader  to  the  end,  and 
leaves  him  with  a  melted  heart  and  moistened  eye.  It  is  a  very  con- 
venient and  attractive  edition.  It  will  contain  all  his  novels,  dramas, 
and  lyrics.  The  latter  have  a  thoroughly  Irish  flavor,  and  will,  we 
sincerely  hope,  be  the  means  of  making  the  talented  young  Irishman 
widely  known,  and,  consequently,  admired  in  this  country. — Putnam's 
Monthly. 

The  works  of  the  author  in  question  are  written  in  an  easy,  flowing 
style,  and  his  sketches  of  character  exhibit  a  rare  knowledge  of  hu- 
man nature — Irish  human  nature  especially — while  the  descriptive 
portions  carry  with  them  a  truthfulness  and  fidelity  which  are  pecu- 
liarly charming.  Many  of  his  narratives  are  founded  upon  real  inci- 
dents in  Irish  history  which  are  ingeniously  worked  up,  and  rendered 
thrillingly  exciting  and  absorbingly  interesting.  They  are  interspers- 
ed with  scenes  of  the  deepest  pathos,  and  the  most  genuine  humor — 
at  one  moment  we  are  convulsed  with  laughter,  at  the  next  affected 
to  tears — while  an  atmosphere  of  sound  moralit}^  containing  many 
touches  of  true  philosophy  and  deep  reasoning,  surrounds  the  whole, 
showing  that  the  author  well  knew  the  way  to  the  human  heart. 
While  every  one  who  possesses  true  literary  taste  will  be  charmed 
with  Griffin's   works,  they  will  be  peculiarly  fascinating 


BOOKS  PUBLISHED  BY  D.  &  J.  SADLIER  &  COMPANY. 

J.  lie  Life  of  our  Lord  and  Saviour  Jesus 
Christ;  or,  JESUS  REVEALED  TO  YOUTH.     By 

the  ^BBE  F.  Lagrange.  Tr^slated  from  the  French  by 
Mrs.  J.  Sadlier  ;  with  the  approbation  of  the  Most  Rev. 
John  Hughes,  D.  D.,  Archbishop  of  New  York.  Illus- 
trated with  10  plates.     16mo. 

.  Cloth  extra,  60  cents ;  Cloth  gilt,  '76  cents. 

"  Suffer  little  children  to  come  unto  me  !  "  said  our  Lord  himself, 
in  the  fullness  of  his  love :  he  wishes,  then  to  draw  their  innocent 
hearts  to  himself,  and  it  is  for  us  to  make  him  known  to  them,  that 
knowing,  they  may  love  him  ;  and  loving,  they  will  be  sure  to  serve 
him  all  their  life  long.  Let  them  behold  him  as  he  was  Avhen,  clothed 
with  our  frail  humanity,  he  walked  amongst  men,  scattering  blessings 
as  he  went.  This  little  volume  is,  I  think,  well  calculated  to  make 
Jesus  known  to  the  younglings  of  his  flock.  It  is  written  in  a  simple, 
conversational  style,  and  contains  a  summary  of  every  thing  relating 
to  our  divine  Saviour  that  children  ought  to  know.  Even  those  who 
are  already  playing  their  allotted  parts  as  men  and  women,  on  the 
world  s  stage,  may  derive  both  pleasure  and  profit  from  perusing  this 
simple  work  of  piety.  It  will  happily  lead  them  back  for  a  time  to 
the  green  pastures  and  the  silvery  waters  of  life's  young  spring,  far 
away  from  the  dusty  highway  and  the  crowded  mart,  and  all  the 
ceaseless  whirl  of  more  recent  years.  They  will  be  children  in  heart, 
at  least  for  the  time ;  and  any  thing  that  assimilates  us  to  children 
must  ever  be  advantageous,  since  Christ  himself  tells  us  that,  unless 
we  become  as  one  of  these  little  ones,  we  shall  have  no  place  in  his 
eternal  kingdom. 

Montreal, 

Feast  of  the  Annunciation  of  B.  V.  Mary, 
March  25th,  1^57. 


T 


lie  Prophecies  of  8§.  Columbkille^ 

MAELTAMLACHT,  ULTAN,  SEADHNA,  COIRE- 
ALL,  BEAROAN,  MALAOHY,  &c.  Together  with  the  pro- 
phetic Collectanea,  or  Gleanings  of  several  writers  who  have  pre- 
served portions  of  the  now  lost  Prophecies  of  our  Saints,  with 
literal  Translation  and  Notes.  By  Nicholas  0 'Kearney.  With 
the  Life  of  ST.  COLUMBKILLE.  By  the  Rev.  Thos.  Walsh. 
16mo.     Cloth,  Price  38  cents. 


Ti 


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Porter.     3  volumes  in  one. 

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JLhe    liife    of  St.    Joseph. 

The  admirable  life  of  the  glorious  patriarch  Saint  Joseph, 

taken  from  the  Cite  Mystique  ;  with  an  Appendix  of  the 

life  of  the  venerable  Maria  of  Jesus,  of  Agreda,  (author 

of  the  Mystical  City  of  God);  and  an  introduction  from 

the  manuscripts  of  M.  Olier,  founder  of  the  Seminaries 

of  St.  Sulpice.     16mo.     Cloth,  extra,  50c. ;  gilt,  75c. 

This  extraordinary  "  Life,"  which  has  been  carefully  extracted  from  the 
Cite  Mystique  (Mystical  City  of  God),  as  it  participates  in  all  the  appro- 
bations accorded  to  the  great  work  from  which  it  is  derived,  by  the  supreme 
Pontiifs,  during  the  two  last  centuries,  cannot  be  too  highly  recommeDded 
to  the  Catholic  reader. 


M 


SUPPLEMENT  TO  **  THE  FOLLOWING  OP  CHRIST." 

ethodieal   and    Explanatory    Table  of 

THE  CHAPTERS  OF  THE  FOLLOWING  OF  CHRIST. 
Arranged  for  every  day  in  the  year  in  the  order  best 
calculated  to  lead  to  perfection.  Compiled  from  the 
Spiritual  Exercises  of  St.  Ignatius  for  the  purpose  of  facil- 
itating the  way  of  giving  Spiritual  Retreats. 

24mo.  Cloth,  extra, 9  c«ots,  or  $6  per  hundred. 

24mo.  Paper 6  cents,  or  $4  per  hundred. 

The  Messrs.  Sadlier  are  happy  to  offer  to  the  Catholic. public,  A  Methodical 
AND  Explanatory  Table  op  the  Chapters  of  the  Following  op  Christ  ;  a 
book  appreciated  not  only  by  the  faithful,  but  also  by  those  separated  from 
the  Church. 

A  certain  method  in  the  reading  of  it  will  be  found  to  render  it  still  more 
useful.  Hence,  a  division  of  chapters  suited  to  the  different  states  of  the 
soul,  to  the  virtues  to  be  acquired,  or  to  the  vices  to  be  corrected,  it  is 
hoped,  will  attain  that  end.  This  division,  by  pointing  off  the  chapters, 
according  to  the  days  of  the  year,  facilitates  the  method  of  meditation 
taught  by  St.  Ignatius,  whilst  it  gives  in  detail  his  directions,  to  those  who 
wish  to  meditate  with  fruit. 

The  order  thus  introduced  into  this  estimable  book  will,  we  hope,  still 
more  enhance  its  ''nlue. 


.  he  liittle  Catechiisiii  5  or,  Questions  and 

ANSWERS  on  those  Truths  which  are  the  most  neces- 
sary for  a  Christian  to  know.  Published  under  the  direc- 
tion of  the  Fathers  of  the  Congregation  of  the  most  Holy 
Redeemer,  and. Missionary  Priests  of  St.  Paul  the  apostle. 

24mo.  Paper  Covers,  $1.50  per  hundred. 


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FIFTH    TJHOUS.lJrn.  ^ 

FROM   THE    SECOND   REVISED   EDITION. 

_KoME— Its  Ruler^  aiifl  its  Institutions. 

By  John  Francis  Maguire,  M.  P. 

At  the  earnest  request  of  several  of  t  je  clergy  and  laity,  we  have  concluded  to 
publish  this  most  important  work.  To  the  Catholic  it  is  invaluable,  and  it  is  just 
the  book  for  those  outside  the  Church.  All  will  be  satisfied  with  what  the  Church 
has  done  and  is  doing  for  the  spiritual  as  well  as  temporal  welfare  of  man. 

It  is  not  necessary  to  say  anything  further,  when  the  highest  authority  in  the 
Church  has  spoken.     We  append  the  following  : — 

Mr.  Maguire  has  been  honored  with  a  letter  from  his  Holiness,  of  which  thefollowxng 
is  a  translation. 
"  Well-beloved  Son  •  Health  and  Apostolical  Benediction — ^We  have  lately  receiv- 
ed your  most  dutiful  letter,  dated  the  17th  day  of  November  last,  in  which  you 
have  TOen  pleased  to  present  to  Us  a  work  composed  by  you  in  the  English  lan- 
guage, and  published  this  year  in  London,  with  the  title,  'Rome,  its  Ruler,  and 
its  Institutions.'  Of  this  work  we  have  been  unable  to  enjoy  the  perusal,  owing  to 
our  extremely  imperfect  acquaintance  with  the  language  in  which  it  is  written. 
Yet,  from  the  statement  of  persons  of  the  highest  character  for  competency  and 
trustworthiness,  who  have  perused  the  same  woi'k.  We  learn  with  peculiar  satis- 
faction that  it  contains  many  evidences  of  your  singular  devotedness.  attachment, 
and  reverence  to  Us,  and  towards  this  Holy  See  ;  a  circumstaYice  which  could  not 
fail  to  be  most  gratifying  to  Our  feelings.  Therefore,  while  bestowing  Our  heart-- 
felt  recommendation  on  this  noble  expression  of  your  sentiments  to  Us,  We  return 
you  thanks  for  the  gift ;  and,  as  a  testimony  of  Our  fatherly  love  towards  you,  We 
impart  to  you  affectionately  from  Our  heart  tte  Apostolic  Benediction. 

Given  at  Rome,  at  St.  Peter's,  the  14th  day  of  December,  1857.  in  the  twelfth 
year  of  Our  Pontificate."  PIUS  IX. 

"  My  Dear  Sir  : —  London,  August  28,  1857. 

According  to  your  desire,  I  have  delayed  acknowledging  the  receipt  of  your 
'  Rome '  till  I  had  read  it  through.  This  I  have  now  done,  taking  it  up  at  every 
leisure  moment  with  renewed  pleasure,  till  I  had  finished  it.  Having  myself  had 
to  go  over  a  great  pai-t  of  the  ground,  whether  personally  or  by  the  study  of  docu- 
ments, I  think  I  am  qualified  to  form  a  just  judgment  of  the  work.  It  is  most 
truthful,  accurate,  and  une^aggerated  picture  of  the  Holy  Father,  of  his  great 
works,  and  of  his  most  noble  and  amiable  character,  drawn  with  freshness,  elegance 
and  vigor — with  admiration,  and  even,  if  you  please,  enthusiasm  ;  but  not  greater 
than  is  sliared  by  every  one  who  has  drawn  near  the  person  of  the  Holy  Father. 
There  is  not  a  trait  in  your  portrait  which  I  do  not  fully  recognize  ;.  not  an  action 
or  a  speech  which  I  could  not  easily  imagine  to  have  been  performed  or  spoken  in 
my  presence— so  like  are  they  to  what  I  have  myself  seen  and  heard.  In  estimat- 
ing what  has  been  done  during  the  late  years  of  quiet  rule  for  the  prosperity  of 
the  Pontifical  States.  I  think  you  have  prudently  kept  rather  below  than  gone 
above  what  might  have  been  stated.  The  result  will  be  more  manifest  in  time — 
to  the  confusion,  one  may  hope,  of  those  who,  dishonestly  or  ignorantly  misrepre- 
sent every  measure  of  the  Sovereign  Pontiff.  I  feel  sure  that  your  work  is  calcu- 
lated to  do  much  good  wherever  it  is  read  ;  and  I  cannot  help  hoping  that  the  very 
novelty  of  daring  to  speak  the  bold  truth,  the  abundance  of  information  which  is 
communicated,  and  the  eloquence  of  the  style,  will  obtain  for  j^our  book  all  the 
popularity  which  it  deserv^.  I  need  not  say  fhat  by  this  work  you  have  nailed 
your  colors  to  the  mast,  and  become  the  Pope's  champion,  in  the  House  as  well  as 
out  of  it  ;  and  I  am  sure  you  Will  not  allow  him  to  be  vilified  by  any  one,  howevei 
lofty.  I  am  ever,  my  dear  sir. 

Your  affectionate  servant  in  Christ, 
John  Francis  Maguire,  Esq.,  M.  P."  N.  Card.  Wiseman. 

Cloth,  extra $1  25 

Half  calf,  or  Morocco 1  75 


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H 


NEW      EDITION. 

istory   of  ihc    l¥ar  in   the   Peninsula 

AI^D  IN  THE  SOUTH  OF  FRANCE.  From  the 
year  1807  to  the  year  1814.  By  VV.  F.  P.  Napier,  C.  B., 
Colonel  H.  P.  Porty-Hiird  Regiment,  Member  of  the 
Royal  Swedish  Academy  of  Military  Sciences. 

Royal  8vo.     800  pages.     Cloth,  extra,  $2.50;'  half  raor.,  $3.00. 

"  The  aiithor*s  style  is  exceedingly  vi^orows  and  terse.  It  has  a  plain 
and  simple  vehemence,  which  is  both  captivating  and  imposing.  He  knows 
nothing  of  amplification.  He  tells  his  story  in  the  fewest  words  possible  ; 
marches  straight  to  a  fact,  seizes  it,  and  places  it  before  the  reader  without 
any  useless  preliminaries,  takes  some  fortress  of  error  built  up  with  care 
and  pains  by  some  precedent,  interested  or  partial  historian,  by  a  coup  de 
rmi'ut.^  and  razes  it  to  the  ground.  His  narrative  is  full  of  life,  motion  and 
impulsiveness ;  his  portraiture  of  character,  bold,  striking  aud  impressive. 
His  honest  and  chivalrous  nature  enabled  him  to  rise  above  the  then  popu- 
lar British  prejudice  against  the  French.  While  he  is  careful  to  guard  the 
military  reputation  of  his  own  countrymen  from  injustice,  he  is  equally  so- 
licitous to  do  justice  to  the  prowess  of  the  French  army,  and  the  military 
skill  of  the  great  men  who  commanded  it.  He  pays  a  proper  tribute  to  the 
military  genius  of  the  great  Napoleon,  whom  he  places  on  the  loftiest  pedes- 
tal in  the  Pantheon  of  the  great  captains  of  .modern  times.  He  loves  to 
paint  him  as  the  great  champion  of  equality  and  the  popular  principle, 
contending  against  the  allied  forces  of  privilege  and  aristocracy." — Tahlet. 

"  On  this  great  work  itself,  we  do  not  jiresume  to  say  much.  The  ablest 
reviewers  in  Great  Britain,  and  Ireland,  and  France,  and  Germany,  have 
concurred  in  deciding  that  it  is  one  of  the  most  valuable  histories  ever  pub- 
lished. The  author  was  a  brave  soldier,  and  a  gentlemen  of  the  finest 
talents  and  training.  He  fought  m  the  wars  he  describes.  Those  ware 
were  the  most  important  of  modern  centuries.  In  them  figured  the  gi-catest 
military  commander  that  ever  lived;  the  fate  of  more  than  one  ancient 
royal  house,  together  with  the  political  condition  of  two  or  three  great  peo- 
ples were  at  stake  in  them;  in  their  conduct  appeared  the  justice,  chicanery 
and  deceit,  the  valor,  and  pusillanimity  of  the  prominent  diplomatists  and 
captains,  and  armies  of  Europe  ;  and  in  their  issxie  appeared,  after  the  waste 
of  millions  of  treasure  and  the  loss  of  tens  of  thousands  of  lives,  the  re- 
settlement of  Southern  Europe  m  its  thrones  and  governments,  and  imper- 
ishable honor  for  the  arms  of  England.  Col.  Napier  has  left  nothing  out 
belonging  to  these  important  facts." — Fitzgerald's  City  Itmn. 


V  he  iHisi^fon  of  Death, 

A  Tale  of  the  New  York  Penal  Laws.  By  M.  E.  Wal- 
woRTR.  1  Vol.  18mo.  Fine  paper,  clo-th  extra,  50  cts. ; 
full  gilt,  75  cents.  Postage,  15  cents. 

X  raetical  Piety, 

Set  forth  by  St.  Francis  of  Sales,  Bishop  and  Prince 
of  Geneva.  Collected  from  his  Letters,  Discourses  and 
Meditations.  Translated  from  the  French.  18 mo.  Cloth 
extra,  50  cents.  •  Postage,  15  Cents. 


BOOKS    PUBLISHED    BY    D.    <fe    J.    SADLIER    <fe    OOMPANT, 


All  Clcnieiitary  4^reek  Crraminar,  based 

OX  THE  LATEST  i>ERMAN  EDITION  OF  iUH- 
NER.     By  Charles  O'Leary,  M.  A. 

Large  12ino.,  half  arabesque,  75  cents. 

The  want  has  been  generally  felt  of  an  Elementary  Greek  Gram- 
mar, on  the  modern  improved  system  of  German  Grammars.  This 
system  has  been  taught  for  some  years  at  Mount  St.  Mary's  College, 
both  by  Professor  O'Leary  and  others,  with  marked  advantage  to'  the 
student  over  the  old  system. 

Induced  by  this  consideration,  Professor  O'Leary  has  compiled  the 
present  treatise.  The  plan  pursued,  is  that  followed  in  teaching  at 
the  above  College,  which  is  virtually  that  adopted  in  the  Grammars 
of  which  it  professes  to  be  an  abridgement. 

The  Grammar  aims  at  giving  in  a  clear  and  brief  form  all  the  ad- 
vantages of  Kiihner's,  apart  from  the  incidental  disquisitions,  numer- 
ous details,  and  special  examples  of  the  original. 

The  student  is  made  acquainted  with  the  laws  that  govern  the  in- 
terchange of  letters,  and  thus  furnished  with  a  key  to  the  forms  and 
changes  of  declinable  words,  which  in  the  old  Grammars  were  mat- 
ters of  rote. 

The  structure  of  the  verb  is  presented  in  an  entirely  new  form, 
whereby  it  is  much  simplified.  The  syntax  is  an  abridgement  of 
Kiihner's,  by  Jelf,  brief  and  simple,  so  as  to  accord  with  the  general 
object  of  the  Grammar. 

"  The  Professor  of  Greek  in  Mount  St.  Mary's  College,  near  Cincin- 
nati, speaks  of  the  order,  clearness,  and  brevity  of  the  new  Grammar 
in  the  highest  terms,  and  thinks  it  destined  to  supersede  all  others 
now  used  in  the  schools." — Catholic  Telegraph. 

'•  The  great  advantage  of  this  Grammar  over  the  Greek  Grammars 
heretofore  in  use  is,  that  it  renders  the  rudiments  of  the  language 
much  easier  to  learn  and  retain,  inasmuch  as  they  are  less  a  multitude 
of  is(»lated  instances  fatiguing  the  memory  of  the  student  thau  the 
legitimate  deductions  of  the  scientific  principles  of  the  language.  We 
recomnieiid  the  book  to  the  examination  of  teachers.  As  far  as  we 
have  had  time  to  examine  it  ourselves,  we  have  been  much  pleased 
with  it." — BrovBiuorCs  Review, 


A 


BOOKS    PUBLISHED    BY    D.    <fe    J.    SADLIER    <fe    COMPANY. 


MERICAN   PRACTICAL.   ARITHIfli:- 

TIC5  (The).  With  New  Aiialj^tical  JMethods  for  the 
Solutions  ofl  Arithrnetical  Problems.  Designed  for  the 
use  of  Pupils  of  t)  e  Bi*^hers  of  the  Christian  Schools. 
By  P.  Ulic  Burke,  M.  D.  E.     380  pages,  12mo.     44  cts. 

The  AMERicAi\  PRirriARir  arith. 

METIC  Designed  for  the  use  of  the- junior  pupils  of 
the  Brothers  of  the  Christian  Schools.  By  P.  Ulic  Burke, 
M.  D.  E.     18nio.,  half-bound.     18  cts. 

Approbation. 

Montreal.  March  30,  1858. 
Sir,— We  have  examined  the  manuscript  copy  of  the  Anthnietic  which  you  scut  us,  and 
think  that  it  is  all  we  could  desire,  and  in  clearness  of  expression  and  reasoning  excels  any 
that  we  have  vet  met  with  in  the  United  States,  and  we  liave  not  the  slightest  objection  to 
your  stating  that  we  have  adopted  it  for  our  schools. 

.  In  saying  th;.t  we  have  adopted  it,  we  have  not  said  sufficient  to  express  our  feelings  ;  I 
should  add  tliatwe  make  this  engagement  most  heartily,  and  witli  sentiments  of  gratitude  to 
the  author,  believing  that  he  has  conferred  a  great  benefit  on  our  Catholic  Schools. 
I  am,  with  profound  respect,  Sir, 

Your  very  humble  and  obedient  Servant, 

F.  FACILE, 
Dr.  P.  U.  Burke,  Utica.  Provincial  of  the  Brothers  of  the  Christian  Schools, 

r; 
Extract  from  the  Pw'ace. 
Treatises  on  Arithmetic  are  so  numeroim^and  excellent,  that  it  might 
seem  superfluous  to  add  another  to  the  number;  yet  it  was  thought  tnat 
room  might  still  be  found  for  one,  which,  discarding,  to  a  great  degree, 
the  multitude  of  complicated  rules  which,  at  the  present  day,  encumber 
this  science,  and  appealing  more  directly  to  the  reasoning  faculties  of  the 
pupils,  might  enable  them  to  solve,  if  not  all,  nearly  all  of  the  most  difficult 
problems  in  arithmetic  by  the  aid  of  the  first  four  fundamental  rules ;  and 
it  is 'hoped,  that  the  reasoning  powers  thus  applied  to  the  discovery  of  the 
relations  and  analogies  found  to  exist  among  numbers,  would  not  cease 
there,  but  would  continue  to  be  applied  by  the  pupils  to  other  pursuits  and 
to  other  circumstances  wholly  foreign  to  calculations,  and  with  results  as 
^tisfactory  as  were  the  solutions  of  the  problems  on  which  they  were  first 
exercised.  It  will,  I  am  sure,  be  also  considered  the  very  best  possible  in- 
troduction to  algebra. 

The  various  properties  of  numbers  present  a  great  variety  of  simple  and 
easy  methods  of  solving  many  problems  which  were,  heretofore,  considered 
to  belong  more  properly  to  the  domain  of  algebra,  but  which  would  require 
considerably  more  time  even  to  put  them  into  the  form  of  an  algebraic 
equation  than  it  takes  to  solve  them  by  reason  and  arithmetical  analysis. 

The  American  Primary  Arithmetic  and  the  American  Pracvical 
Arithmetic. — Both  these  works  are  the  production  of  P.  U.  Burke, 
M.  D.  E.,  and  both  are  published  by  the  Sadlicrs.  They  are  among  the  verj 
best  Arithmetics  we  have  seen.  At  first  we  were  disposed  to  condemn  the 
too  technical  acci.iracy  of  the  introductory  book  ;  but  when  we»  consider 
that  the  exact  scie'nces  dertiand  the  utmost  exactness,  we  believe  it  is  best 
to  famjliarize  the4eamicr  as  early ^^s  possible  with  precise  scientific  terms. 
The  book  rightly  prepared  cannot  be  too  technical.  The  larger  work,  the 
Practical  Arithmetic,  is  far  more  than  its  modest  author  claims  for  it  It 
is  not  only  an  improved  treatise  on  its  subject,  but  an  improved  introduc- 
tion to  all  analyticJil  study. —  Catholic  Mirrtr. 


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'VCY  o/vj>  ^7 


911405 

THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


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BOSTON,  123"^^^^]?  AI.  STRBBT, 
MOJ TREAL,  Comer  of  Kctro  Dame  rnd  St.  Trancis  Xavior  Streets, 
MAV    for  srilea  large  stock  of  OAT  !    Jl  IC  WO!;KS  011  terms  more 
:;ivorab!e  than  ariv^oitif^r  esi  ■'  ii-hn     1.  in  i):;  coaitry.     Theii*  sui'ply  ] 
of  SriIO(-)Ij   JJO<)IlS  of,  heir  own  uuhWanicu.  and   others  is  un- 
usually  large,    '.id  th. ',  a;c  },ir  purc.l  hi  m!)  i'  .;es.  to  :iuj)i>'y  ordcrr^  from 
the  Most  i'"^.  Archlis'Aopi,,  R\i  '  t  Uev.  iii^  ..op.  ,  K«'V    '.  "-le  TrVj  h'fligi- 
ous  (-'OmmiMiUiee,  and  tho  pn'^iii,  -n  geri»>rat  vvi^h  prompt  attc-ntion. 

Among  their  valuable  pn.  /icijtions  are  tli"  foilowin,^  ScIkjo.  liooks,  j 
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